When it comes to matching components for vinyl playback, there are two things you need to worry about: matching the cartridge to the tonearm and matching the phono stage to the cartridge. Other than that, a good amp is a good amp regardless of the source.
When matching the cartridge to the tonearm, a heavy tonearm requires a low compliance cartridge and a low mass tonearm requires a high compliance cartridge. Most current tonearms are medium mass and most current cartridges are medium compliance. You actually have to try to get a mismatch with current new products, but it is certainly possible. There are formulas for calculating the resonant frequency of the cartridge/tonearm system, but you need the actual dynamic compliance of the cartridge at 10Hz and the effective mass of the tonearm for those calculations to be accurate. Those specs can be difficult to find. As a rule of thumb, don't put a heavy tracking cartridge (VTF => 3g) on a thin, ultralight straight tonearm and don't put a light tracking cartridge (VTF <= 1.5g) on a heavy broadcast arm. Once you get into vintage cartridges and tonearms, it becomes pretty easy to get a mismatch. When in doubt, do your research. A google search for recommended cartridges for your turntable/tonearm will usually get you pointed in the right direction.
For cartridge/phono stage matching, whether it's the internal phono stage of an integrated amp or a stand alone external phono stage, it's important to match the input impedance and capacitance of the phono stage to the cartridge manufacturer's recommendations. For most current cartridges, these specs are easy to obtain, they will usually say something like 47K ohms and 150 - 300pF. The exact values vary from cart to cart. 47K is pretty standard, especially for current products, and most phono stages are set to that value (or very close) by default. Some phono stages offer other settings, and some don't. Same for input capacitance. Some have a fixed value, others allow multiple settings. It can be difficult to find the input capacitance for many current home theater amplifiers that include a phono stage. That's usually a sign it's higher than it should be and adding a phono input was an afterthought (i.e. it'll probably work, but probably not sound great). In this paragraph, I'm referring specifically to MM/MI/HOMC cartridges. LOMC get a little more complicated, but most beginners don't start out with a LOMC cartridge. If you do, again, google is your friend.
Edit: this post isn't meant to provide all the answers, just a view from 30,000 feet of the basic issues.