One of the amazing things about vinyl records is that physical dimensions, rotational speeds, and equalization standards (with a few exceptions) have been fixed for a while now. This is great because turntables, or more precisely, record players (turntable, tonearm and cartridge) built a while ago can still be used. It also worth considering that buying a new turntable is a potentially long term purchase that you will enjoy for many years. So buy a good one!
Once you have a ball park budget in mind (which might also need to include a corresponding phono stage of similar performance level) the next decision is are you looking more for a no-fuss plug-and-play kind of record player that doesn't need much attention or maintenance, or are you looking to configure, tweak, and upgrade?
Many enjoy the collectability of vinyl records and may already have, or plan on acquiring, a sizeable vinyl record collection. The turntable should match these aspirations since a better record player will preserve the sound quality of your vinyl record collection investment.
Like a HiFi system, in general you will get the best sound quality for your money if the three are balanced. But if you are planning to evolve a system over time, it is probably better to focus on the table first. The turntable has a bigger role than rotating a record at a constant speed but this is a starting point. More expensive tables are accompanied by an electronic speed controller and these can become quite sophisticated to ensure the motor is provided with very stable and distortion free power.
The bearing supporting the platter needs to be exceptionally smooth with no noise or play. The tonearm mounting system needs to ensure that there is no play or relative motion to the main beaing. The design of the platter, plinth, motor mount, bearing, and tonearm mount forms the foundation from which everything else is referenced. These are all closely coupled together and subject to mechanical vibration energy that interacts with the vibration of the stylus set in motion from the groove of the rotating record. Rigidity prevents resonances from developing, which in a vibration measuring system, would add colorations and degrade resolution. Tables don't have to be large and massive to perform well. Mass tends to convert higher frequency resonances into lower frequency ones and may store and subsequently release energy or redirect it away. This is why what you put a record player on also influences the sound.
The tonearm is a separate element from the table. There are companies that just make tonearms or tables. Among those companies only manufacturing tables, options and ease of installation of different tonearms varies. Tonearms also need to be very rigid, stable, and ideally with complete movement in two degrees of freedom; vertically and horizontally. Vibrational x or y axis translation or z-axis rotation, bending, or regional resonances need to be as minimal as possible. Low friction without complete stability is not enough. Detachable headshells and real-time user adjustmens such as tonearm height, unless very well engineered, may introduce resonances by compromising ridigity. Because the voltage signal output from cartridges is tiny, tonearm leads make as much of a difference, if not more, than interconnects.
The table and the tonearm quality determine the range of performance achievable with more expensive cartridges. An inexpensive cartridge can sound better than you might expect fitted to an expensive turntable/tonearm, similarly you may be disappointed with an expensive cartridge in a modest record player. Cartridges must be installed correctly so the alignment is not off for best performance.
kemela has brands representing plug and play and à la carte approaches; it is for the customer to decide which suits them best. Modern manufacturing has vastly improved the performance of turntables, tonearms and cartridges. There are far fewer badly pressed records and less recycled vinyl.
Turntables are visual and hands on components but at the end of the day it's enjoying the record that counts - and keeping them clean! Also check out the new vinyl record orientated magazines now out on the shelves; they fill in the history behind the bands with stimulating articles about collecting vinyl records.