Your panel should have information printed on it about what breakers are acceptable. My Siemens main panel / meter base has a big sticker on the inside of the door with a lot of useful info. For example, it says the left side can only take 50A maximum with aluminum conductors, while the right side can go to 150A.
However, while the panel maker only recommends its own-branded breakers in its panel, there are 3rd-party manufacturers, like Connecticut Electric that make compatible hardware. In my experience, they work fine but the quality isn't as good. For example, while torquing screws on a 3rd-party ground bar to the Siemens-recommended level (35 in lb), I destroyed the screw.
The easiest way to get a matching breaker is to pop one out and take it to the store with you. It's a good idea to turn off the main breaker while you do this.
I've had a very difficult time getting exactly the right electrical components at hardware stores. If there's an electrical supply house in your area, you may have more luck there.
14ga copper conductors are normally the minimum for a 15A circuit, and 15A is plenty for most computers, unless you run a rack of servers. Many electricians use 12ga for both 15A and 20A circuits, since it's easier to stock just one size of wire. If you do that, you may as well stick with your plan of a 20A circuit to your computer area.
Minimum conductor size is determined by ampacity - the amount of current you can safely draw without worrying about the wire overheating. However, over long runs, the resistance of the wire becomes a factor, causing voltage drop. The amount of voltage drop depends on the size of the wire, copper vs. aluminum, stranded vs. solid, the length of the run, the load, and heat. As current increases, voltage drops (V=IR).
Note that we're talking about actual load, not rated load. That is, if a computer is plugged in to a 20A circuit but only drawing 5A, the voltage drop is based on the 5A.
Voltage drop is particularly concerning for motors, especially when they start, as the starting current is extremely high. If you are running large saws in your workshop, you can protect the motors by oversizing the conductors.
If both circuits are running to the same location, consider using a multi-wire branch circuit (MWBC). You run 4 conductors (hot/hot/neutral/ground), and treated it like two separate 3-conductor circuits, so you use much less copper. You can just buy a roll of 12/3 + G cable and run it from the panel to all your receptacles, which is pretty convenient. If the loads are well-balanced, it can also reduce voltage drop.
Between either hot and the neutral gives 120V. Between the two hots gives you 240V, so make sure the breaker is off when working on it. You'll need to use a 2-pole breaker (not two separate 1-pole breakers) so that both hots get shut off at the same time.
You can even wire the top & bottom of each receptacle to be on different sides of the MWBC, so it's easy to balance out the load. That's the goal: get half the load on each side of the circuit, as well as you can. The recep will have two screws on each side and a tab you can break off to separate them - do that on the hot side.
