I am trying to find out if I can use a 14/3 cable to power two separate circuits, and to do that I know I must use two separate phases, but I am not sure how to do that.
What you're talking about is a Multi-Wire Branch Circuit. There are two problems with you trying to take one of these on.
- They are effectively obsolete, as GFCIs and AFCIs are now required on the vast majority of 120V circuits, and MWBCs do not play well with GFCIs and AFCIs.
- They are for wizards only: You need to know many more things to do an MWBC safely, and there are many ways to mess up and create a dangerous situation. Yes, at the very least a 2-pole breaker is required. A classic blunder is going "Gee, a double-stuff has 2 breakers/screws, so it'll let me put an MWBC in 1 space". No, it won't - this will set your neutral wire on fire. These are the kinds of pitfalls that exist with MWBCs.
Given the difficulty of doing these circuits safely, and the fact that you probably can't even use them due to GFCI/AFCI requirements -- I would say that they are an obsolete/deprecated method, and advise you to simply run two 14/2's.
The technique is allowed, you can google "MWBC" but I'd spend at least several hours "learning up" on the various gory details and what those mean.
Even if you do the MWBC textbook-perfect, you could find at home sale time, your AHJ sweeps in and says "Where are your AFCI breakers?" and "Where are your GFCIs?" And you find you are now painted into a corner and have to rewire just to sell your house.
It is allowed (in my locality), but you must have a handle tie on the circuit breakers. The neutral will be shared between the two circuits. You must make sure that red and black wires have the voltage 180 degrees apart, or else you can overload the neutral current. Also, check your local code about the usage of arc-fault protection... it is now required almost everywhere here. You can accomplish that with a special circuit breaker, but it must be a dual pole type for your purpose (will be expensive).