There are many different options for buying a violin, which include:
Music Shop - a general music retailer, offering a wide range of products for many different types of musical instrument.
Violin (String) Shop - a specialist music retailer, usually with in-house luthiers (violin repairer/maker), who have specific expertise in violins and violas (and usually cellos and double basses too).
Violin Dealer - a high-end specialist version of a Violin Shop (usually including luthiers). Dealers tend to trade from premium showrooms in expensive areas, or from their homes, don't stock beginner-level instruments, and mainly handle instruments worth tens or hundreds of thousands of dollars!
Online Store - an online version of either a music shop or a violin/string shop. Sometimes these are online-only stores, sometimes they are part of a retailer who also has an offline presence.
Private Seller - you can often find individual people wanting to buy and sell instruments, such as committed non-professional players who will advertise through local organisations and contacts, or by word of mouth.
The setup of a violin is crucial. This means making sure that it is correctly fitted out with the bridge, strings, pegs, fine tuners, and soundpost all in the correct positions and securely checked over by a good craftsman. For this reason, for learners just starting out, we would always recommend a violin shop as the best place to start. This allows you to be sure that an experienced professional has prepared your instrument, and that it is in good condition and won't cause you any problems in the first few weeks (apart from getting it in tune! more on that in a future lesson...!)
If you order a violin online, you might find that it comes without the bridge being set up properly, and that you still need to take it to a music shop to have it fitted out properly. Another problem is that many music shops are not string specialists, so although they will be happy to sell (or sometimes rent) a violin to you, they often won't have either the specialist in-house luthier services required, to be sure that the setup is right.
Also, without a sales specialist who understand the needs of violin players, you might end up being sold the wrong kind of chin rest or shoulder rest for your needs. So if you do choose to buy from a non-specialist retailer, make sure you are well-informed and that you know exactly what you are buying!
There are some retail channels that you should avoid completely. In the UK, where ViolinSchool's HQ is based, we often see instruments that people have bought online from Tesco (a major supermarket), and Argos (a catalogue-based high street / online retailer). These violins are cheap beginner-level instruments supplied with absolutely no specialist setup whatsoever, and they come with the worst possible quality strings, poor craftsmanship, and inferior-quality accessories.
More often than not, we find that learners choosing the very cheapest option will almost always get frustrated by the lack of quality, and either become demotivated because the violin doesn't produce the sounds they want, or have to then get a replacement violin (and end up spending more altogether). So don't make that mistake... at the cheapest end of the market, you really do get what you pay for!