There are several good choices near the center of town. One of the best budget options is the Pousada Ecoverde (Rua Pedero Celestino 391, tel. 65/3624-1386, www.ecoverdetours.com [1], R$25 pp). Operated by local English-speaking eco-guide, Joel Souza (see Getting Around), the simple rooms (two with shared bathrooms) are located in a charming old house belonging to Souza’s family. You can borrow books from the library and retire to a hammock in the gardens shaded by fruit trees. Laundry and cooking facilities are available.
More upscale and comfortable is the Amazon Plaza (Av. Getúlio Vargas 600, tel. 65/2121-2000, www.hotelamazon.com.br [2], R$132–168), one of the nicer modern hotels in the center. Although the large, nondescript rooms are blessedly air-conditioned, you can also cool off in the small pool, surrounded by a jungly garden. The restaurant features great breakfasts as well as regional fare.
Cuiabá [3] has a good range of restaurants, most of them very affordable. One of the favorite local traditions is to eat at a peixaria, a typical eatery that serves fresh river fish such as piraputanga, pintado, and pacu. Peixaria Popular (Av. São Sebastião 2324, tel. 65/3322-5471, 11 a.m.–3 p.m. and 7 p.m.–midnight Mon.–Sat., 11 a.m.–5 p.m. Sun., R$12–25) is a perennial favorite, where you can try the Mato Grossense specialty known as mojica de pintado. The generous portions easily feed two.
Further from the center in the bairro of Araés is 
Biba’s Peixaria (Rua General João Severiano da Fonseca 508, tel. 65/3322-3174, 11 a.m.–3 p.m. and 7–10:30 p.m. Mon.–Fri., 11 a.m.–4 p.m. Sat.–Sun., R$20–35). Tucked away in a little hidden street, Biba’s is the place to get fresh Pantanal fish. Choose from à la carte specialties such as pacu accompanied by farofa (toasted manioc flour) with banana, rice, and salad, or indulge in the rodízio, which will allow you to sample a little bit of everything. For dessert, try the rapadura de leite, a fudge-like doce that is typically Cuiabana.
Another good place for local cuisine is O Regionalíssimo (Av. Beira Rio, tel. 65/3623-6881, 11 a.m.–2:30 p.m. Tues.–Sun.). Located next to the Museu do Rio [4], this self-service buffet specializes in local fare. Although fish is the mainstay, there are also meat and vegetable dishes. Afterwards, head to Alaska (Av. Isaac Póvoas 1100, tel. 65/3028-2723), a local sorveteria chain, for some sweet respite from the heat.
Links:
[1] http://www.ecoverdetours.com
[2] http://www.hotelamazon.com.br
[3] http://www.moon.com/destinations/brazil/brasilia-and-the-pantanal/the-pantanal/mato-grosso/cuiaba
[4] http://www.moon.com/destinations/brazil/brasilia-and-the-pantanal/the-pantanal/mato-grosso/cuiaba/sights