I've been to Krewe de Vieux a couple of times recently. Once I watched from the Elysian Fields neutral ground area; I liked that spot. The other time, we were were at Kerelec St; that was much more crowded and cramped, but had the advantage of the R Bar right there for beverages & bathrooms (as they say, "ain't no where to pee on Mardi Gras Day")... Of course, New Orleans lenient alcohol laws mean you can save money and BYO beverage to the parade, in cups or cans, no glass.
I think a bike is a great way to get around the city, just look out for bad drivers and big potholes. Bike rentals can be expensive, around $30/day, but see if your Air BnB host can loan you one.
The streetcars are a good way to get around, and the buses are too, although they can run infrequently and not always on schedule. Info at norta.com.
If you liked Treme, I assume you're interested in music. Frenchmen St is a must see. The clubs there mostly have low cover charges, and some are free, including Spotted Cat (where many scenes were filmed). It'll be crazy packed on KdV night, so check it out another night too. Check offbeat.com in early February for complete music listings.
The Canal ferry, which played a memorable part in season one, is still free, though there may be a $2 fare by the time you get to town. It's a short ride but a great way to see the "good side of the city", the view from the River.
Quality meals don't have to mean expensive, there are a lot of great cheap eats in town. Since you're stating in the Bywater, try Elizabeth's for brunch. If you do want a splurge sort of meal, consider lunch, which can be a lot cheaper than dinner at many of the nicer spots in town.