In general, once I settle on an instrument and a set of strings with characteristics that I like (and break them in a bit), the task begins! First off, it needs to be said that I use a relatively full-range, flat-ish response cab, and I have my amp dialed in pretty much dead flat (as a starting position) - which leaves most of my tone-shaping to come from my Tech-21 VTDI.
By default, my VTDI includes a speaker sim that I like, and with that feature engaged as a baseline, my first goal is to dial in drive. I like my bass to sound like an electric instrument - and part of what that means to me, is just a bit of distortion audible on hard attacks or double stops, and an overall less perceptible warmth and subtle compression. I feel like this distortions helps bass be audible in a mix without being overpowering, and it also aids in expression - if I dig in, it sounds loud without actually being loud and blowing up any meters.
Next is EQ. Now, I understand that this process won't work for everybody, but given my rig notes above (flat amp and FRFR-ish cab), my method for finding a good starting point is actually to record some bass against a few of the bass-less tracks from one of my bands that I have on hand. I'll then listen to the recordings on a few different systems and against reference material (just like when mixing) and will tweak as necessary. That's initially how I figure out how a new bass/strings combination will perform in a mix.
A this point some of you might be raising an eyebrow, thinking that a recorded mix isn't the same as a live mix - and while technically you'd be right, I've yet to find a recording tone that works well for me, that also doesn't also work live, at least as a starting point.
From there, on stage, I can tweak my amp from flat as necessary to make any minor adjustments. That's less about dialing in my tone, and more just trying to adjust to the room though - no method to that, I just play it by ear. If something is bothering me, then I'll adjust my settings.
Aside from regular minor adjustments over the course of 3-6 months (usually), that's all there is to it!