I'm looking for fine grain street photos with a pretty high contrast and a nice density. I've heard various things about pull processing and push processing Tri-X 400 film, so I'm a little confused.

Tri X is pretty verstial, it can be pushed to 3200, but can become grainy. Tmax 400 can be pushed to 3200 as well and Tmax 3200 can be pushed to 6400, I think grain is better with Tmax 3200 shot at 1200 to 1600 than Tri X pushed. I only use Tmax developer or DDX with Tmax or Delta, for Tri X, HP 5, or Forma 400 I use Edwal 12, Microdal X 1:3, there are lots of combos each with its own look. If I am after an older 70s look I use Forma 400 and Microdal X. Many Tri X shooter like Xtol or HC 110.

Also, I'm wondering if I should switch from TMax Dev to D-76, as I hear it yields better results.

The only way to find out to try both and see what you like best.

If I do switch to D-76 developer, what are the times for push processing/pull processing.

Kodak data guide gives times and temps for TriX and D76, as does the massive development chart.