How much should I plan for a month of backpacking in Taiwan?

If you have a whole month you can really explore many different parts of the island. I suggest doing some research so you have some ideas about places to visit before you come. At the same time be flexible as you will undoubtedly find there are places you might like to visit that you haven't read about. You also have enough time to visit some of Taiwan's smaller islands.

There is quite a lot of information about places to visit in and around Taipei on this site. However, I recommend purchasing a good guidebook. Lonely Planet is probably the most practical and useful.

Is 3 days for Taipei enough?

I assume you will fly into Taipei. Spend a few days there after you arrive. Then plan for a few more days later in your visit. The second time you will probably have a different perspective of the city.

I need a recommendation for a cheap place to stay in Taipei.

See the budget accommodation in Taipei section of David's Guide to Taiwan here at Taiwanderful.

Should I go to Tainan at the south?

Absolutely. Tainan is one of the highlights of Taiwan. The city has so many interesting attractions and you can experience a lot of Taiwan's culture and history there.