Most foreigners to Brazil/Floripa (me included, I'm dutch, but lived in Floripa for 7 years) simply love Lagoa da Conceição. It has a very central position on the island, has enough infrastructure to simply get what you need and it's right next to the beautiful lake/lagoon which is great for walking, kite-/windsurfing, etc. Also from there you can easily get to Joaquina beach (it's a great walk through the dunes), "Praia Mole" and "Barra da Lagoa", each with it's own unique atmosphere.
Be aware that Lagoa da Conceição is divided in the areas:
- Centrinho da Lagoa;
- Porto da Lagoa;
- Canto da Lagoa.
You'll probably want to stay in the first one, where you have everything within walking distance, for the other areas you'd probably need a car, or at least a bike. It's definitely not the cheapest area, but I think it's well worth the money, you will save on transport costs and will get the best experience.
For public transportation info I think Moovit is the best app. Also I believe they just started Uber in Floripa, which might even be cheaper than public transport in some cases (especially when travelling with friends).