The easiest in terms of pure transit is the corridor in Midtown along the Red Line. From about Alabama up to Downtown. Second is Montrose which for some people will have a cooler vibe.
Within Midtown, the further North you go, there are a few more bars/clubs that make noise at night, so either live further South, live a few blocks East or West of the rail, or both. Huge amounts of apartment stock just came online in the last 3 years there, so you might get a solid rent deal somewhere? For me personally, being able to walk to a grocery store is always key anywhere in the world: shop every 2-3 days on the way home from work, only buy what I need, less waste. That area has two grocery stores: Randall's and Whole Foods.
Montrose is the other top spot. For me it is the #1 example of how Houston's lack of zoning can shine (versus other areas where it is a detriment). Fantastic local retail interspersed among every-other residential block. Montrose is a large-ish area and every half mile can be a slightly different "feel", but there are great options to rent newer apartments, older apartments, or rent spacious, cheap, historical houses close to things like Menil Park, or the big restaurant scene along Westheimer. Sadly the Kroger at Hawthorne is closing, but the huge HEB is dead-center of Montrose. East a bit and you're near Trader Joe's. If you're anywhere along Westheimer, the 82 bus goes to to Midtown Whole Foods and if you're along West Gray the 32 bus goes to the Kroger near River Oaks. The whole area always has things happening (check the pinboards at Inversion or Black Hole), and is next door to Midtown for even more.
If Downtown would get a larger grocery store with slightly more affordable staples (no offense to Phoenecia), and have it close-ish to where the rail lines meet, the entirety of Downtown would immediately become more viable (imo). I have friends both in Downtown, and in the Heights (solid bike culture there), but they eat out more than I do. The Heights started getting expensive ten years ago and doesn't look to stop, but it's still an option.
In about 5 years I think the Greater East End will be another great walkable/bikeable region (the actual East End, not EaDo), so depending where along the gentrification curve you prefer to live (before, during, after) it might be worth checking out now.