This is one heck of a mine field you asked here. lol
Like film, this really depends on the budget of the piece/series. In my personal case, the production company I work for has two audio engineers. My coworker composes music for the shows. That's all he does. I do everything else. For the most part, I try to treat each episode or one off that we do similar to a film mix.
I always start with dialogue. For television, as with most other mediums, the dialogue is the most important element. So, that has to be done first; and in a manner that gives it the attention it requires. From there, I typically move on to ambiences. These help to smooth out and settle the dialogue into the picture. So, once again, here the dialogue is king.
After those two elements have been done, I'm usually starting to see the back end of the project approaching. Now it's a war of attrition. What can I get done in the amount of time left? This usually means that I'll make two passes at sound effects. I will spot all critical sound effects required for the story, on screen actions, etc...get those edited and placed. Then I will devote whatever time I have left prior to the start of the mix filling out the effects tracks, trying to imbue a greater sense of perspective to the story, augmenting scenes beyond just the critical needs of the shot.
This is about the time that I get the music from the other audio engineer. My mixes always begin with the dialogue. In television, unlike music (and to a certain degree, film) there are rules as to where the dialogue levels should be. This "Dialnorm" varies depending on the network you're producing for. I get the dialog even and smooth, and try to impose as much dynamic range as is allowed within the network spec. Once I'm happy with the levels, it's on to the ambiences and backgrounds, which are much easier to set once the dialog premix is done. I'll move on to mixing the effects then, but really, the levels you've already set tend to dictate where the effects will sit in the mix. You get a good sense of where they should be when mixing the dialogue and ambiences; it's just a matter of getting them to the proper spot in the mix.
Music comes to me pre-mixed basically in reference to the dialog. My coworker knows essentially where the dialog is going to be level-wise (because of network specs), and has the raw dialog as reference. I bring all of these premixes into a new session, and do my final mix there. From there, it's just approvals/tweaks/etc. as necessary or until we hit deadline. Then layback is done and it's on to the next.
This is just our procedure, but I imagine it's similar to a lot of other peoples'. No matter what though, it always starts with dialogue. Hopefully someone else is willing to share their workflow with you so you can get an impression of the broad range of types that exist.