The following recommendations concern lower priced beginner guitars not guitars for advanced players. All prices are approximate in Canadian dollars. I am not affiliated in any way with any of the companies listed.
It is not necessary to spend a lot of money on a guitar that your child will outgrow. The most important thing is to buy a guitar that is the right size for them and easy to play. Montana makes a variety of small guitars which retail for about $100 including half size guitars which are ideal for very young children (5-9 years old). An instrument at this price is obviously not the highest quality instrument available but they are well constructed and play well. Some Montana guitars are made in Romania and some in China and the Romanian ones are a bit better quality.
Yamaha offers a line of smaller sized student guitars that are of better quality than the Montanas in the $120-$140 range. The CGS102A is a 1/2 sized guitar while the CGS103A is a 3/4 sized guitar. They also have a 7/8 sized guitar called the CS40.
For a guitar of even better quality though also more expensive, there are two options from Godin. The first is the Ami under the Art and Lutherie brand which has a smaller body and a 7/8 size neck. This comes in both a steel string and nylon string model. There is also the Motif in the LaPatrie classical line which has a regular sized neck but a smaller body, i.e. good for a child whose arms are long enough to play a regular sized neck but whose body is still too small to sit comfortably with an adult sized guitar.
LaPatrie line of guitars: Motif (parlour guitar with a small body ideal for child or very small adult), Etude, Concert, Presentation and Collection in order of cost/quality. The price range of these guitar is from $300 to $500 with the Motif being below $300. All of them have a solid wood top (very important) and the top of the line Collection is all solid wood. Options available on some of the models include a cutaway version and electronic system allowing amplification both of which add to the price. These guitars are easy to play and sound excellent for the price, far better than most other brands in this price range. The thin lacquer finish is good for the sound but does make them a bit more delicate. If you are going to take care of your guitar this is an excellent choice but if you are going to be leaving it around and bringing it to parties and campfires it may not be the best choice.
Yamaha guitars is manufacturing guitars with solid tops at a lower price range than they used to and now is a solid competitor to LaPatrie. For those on a tight budget I would recommend the C40 ($140) or the CG102 ($200). Neither of these budget guitars is particularly good but they are solidly made and not difficult to play so adequate for learning on if you can not afford a better guitar. Yamaha actually offers a full sized solid top guitar at a lower price point than LaPatrie with its CG122MS and CG122MC models. The following models are competitive with the LaPatrie models of a similar price CG142C/CG142S (competes with LaPatrie Etude), CG162C/CG162S (Concert), CG182C/CG182S (Presentation) and the CG192C/CG192S (Collection). As the guitars do have a different sound and feel than their La Patrie equivalent it is good if you can find a store to compare them side by side to find which one you prefer. The C and S at the end of the model numbers refers to whether the guitar top is made from cedar or spruce. LaPatries are all made of cedar tops so those players who like the sound of spruce top guitars may prefer to get a Yamaha S model instead of a LaPatrie.
Steel String Guitars
Unless you are tall or plan to play standing up most of the time, I recommend smaller steel-string guitar body styles such as Folk or OM (Orchestra Model) over the larger Dreadnought and Jumbo styles. I have had many students who have had difficulty playing guitars that are too large for their body and even those who have not had difficulty often tell me they wish they had bought a smaller sized guitar. The smaller body guitars also tend to have a more beautiful, refined sound but are not as loud as the larger ones. If you do prefer the larger styles try the dreadnought or jumbo guitar models made by the brands I recommend below.
The maker of La Patrie, Godin, also produces 4 steel-string guitar lines, Norman, Simon & Patrick, Seagull and Art & Lutherie. Each line has some minor differences in construction style. At the lower price range of roughly $300, there are two models, the Simon and Patrick Songsmith Folk which has a solid Spruce top and the Art & Lutherie Folk which is available in cedar or spruce. Around $400 is the spruce top Norman B20 Folk and the cedar top Seagull Coastline S6 Folk (this guitar like many though not all of the Seagull guitars also has a slightly wider neck which is good for people with big fingers). At the $500 range there is the Norman ST40 Folk which is a solid cedar top as well as having solid Mahogany back and sides (the previous lower priced guitars all have laminate back and sides).
Yamaha starts selling solid top steel-string guitars at around the $200 range beginning with the spruce top FG700S. However, this is the larger dreadnought style and they do not have a smaller bodied steel-string at this price level. At the $300 level there is a smaller bodied guitar with a solid spruce top, the FS720. Around $500 they have the LS6 which I find a bit of a high price for a guitar with laminate sides and tops unlike the similarly priced Norman ST40 which is all solid wood. The LS16 is all solid wood but is in the $600 price range.
The greatest amount of competition is in the electric guitar market with the number of brands and models of guitars too numerous to list. I will recommend a few brands and models but don't be afraid to try some other models of the same brand at a similar or higher price to the ones I recommend. However, the lower priced models of the same brands may not be of the same quality as the instruments I am recommending.
A major problem with budget level electric guitars is tuning. While cheap classical and acoustic guitars do not usually go out of tune many cheap electric guitars constantly go out of tune. This is very frustrating and I recommend paying a bit more (roughly $400 and up) for a beginning electric guitar than for an acoustic for this reason. I also recommend staying away from guitars with a whammy bar until at least the $500 range and even then do some research and look at reviews to see if the whammy system of a guitar is good. Whammy bars are notorious for putting guitars out of tune. There are good whammy bar/locking nut systems that can keep the guitar in tune but they are not found on budget guitars.
The lowest priced (a bit under $400) electric guitar I would recommend is the Ibanez RG 321/RGR321 series. These guitars have a fixed bridge not a whammy bar and are quite well made for the price. Ibanez also makes the same guitars with whammy bars called the RG350 line and they are only slightly higher priced than the 321s. However, at this price range I think a fixed bridge is a better bet than a whammy bar system of mediocre quality so I would recommend the 321s over the 350s.
At the $500 range there are a number of guitars that are a good value for the price and the whammy bar systems start getting more reliable. For an all-round guitar good for many styles I would recommend the Godin Session series or the PRS SE (Student Edition) series. Those who want a guitar with more of a heavy metal sound might prefer a Jackson Soloist. For fans of either the Fender Stratocaster or Telecaster sound I would recommend skipping the cheaper, poorer quality Squier line and making the $500 investment to get one of Fender's mid priced line of Mexican made Stratocasters or Telecasters. While I am not a fan of the lower priced Epiphones, once one gets around the $500 range and higher they start to become decent guitars. Thus, fans of the Gibson Les Paul or SG sound could buy one of the better Epiphone models of these guitars.
There are a number of good, small and cheap practice amps on the market today. While one can not expect the same quality of sound as a larger more expensive amplifier, many of the today's practice amps sound surprisingly good. Most include several types of built in effects (digital processing that alters the sound of the guitar) and some are also modelling amps. Modelling amps are amps that use digital processing to imitate the amp sound of various other amplifier models. When this is done well it means you can produce a great variety of sounds on one amp.
The Roland Cube series of amplifiers are a good budget level line of practice amps that are modelling amps as well as featuring a number of other sound effects. This line also includes one very small portable amp, the Micro Cube , that can run on either batteries or a regular outlet. It has the same quality of sound as the other Roland Cube series amps but sacrifices some volume for the ability to be run using batteries. If the capability to run on batteries is useful for you then buy the Micro Cube, if not buy one of the other amps in this line. Vox also produces a fine line of modelling amps such as the VT20 Plus which retails around $180. The Vox line is very competitive with the Roland so try them together to decide which you prefer.
Other manufactures eschew producing modelling amps and concentrate more on just producing one particular type of sound well. Most of these non-modelling amps will still have some effects. The Marshall MG series is an example of this type of amp and is available in various wattages (the higher the wattage the louder and more expensive the amp) and with various effects. While not a tube amp like the more expensive amps made by Marshall it has a technology to try and reproduce the tube amp sound that made Marshall amps famous. Peavey also has a series of smaller transistor amps that try to imitate the sound of tube amps called the TransTube series with models such as the Envoy, Rage and Bandit.
The question is should you choose a modelling amp or a non-modelling amp? I think a valid argument can be made that the non-modelling amps like the Marshall and Peavey amps do produce one sound a bit better than the modelling amps that produce many sounds good but not one sound great. On the other hand even though it is impressive how good these lower priced amps sound, there is a limit to the quality of sound you will get with amps at this price range. My personal recommendation at this price range would be to go with a modelling amp because they produce a lot of sounds quite well and you can have a lot of fun with different sounds and effect. I would wait until you have the money to buy a more powerful and expensive amp to find one that produces the sound you want at a very high quality. Also by this time you will have a better idea of the sound you like. However, I also do not want to discourage people from trying the non-modelling amps I recommend (as well as trying those I have not recommended, there are a lot of brands so there are both some good and poor amps/guitars that I have not mentioned). Some people have a particular sound they like and if a non-modelling amp produces that sound better than a modelling one do not be afraid to choose it over a modelling amp.