If you have finance background, it might be easy for you to get into BA. You just need to have learn about business processes and standards and learn as you work. Some of the QA jobs can be no brainer like manual or black box testing. Of course as a starter, you can join dhoti companies who will train you and help you with jobs.
My first job out of school was database developer and I learnt all at work. As @sojhoketa dai suggested, you have got good resources online now a days. It can get messy when you start playing with big database with complex queries. So its very important to get the concept and start in a right way.
Programming - There are several languages so u might wanna find out which you want to learn - java, .net, php, unix. Oh yeah, Unix scripting has a lot of scope too esp in support. And there's many more you can do in IT world like networking/ data center/IT security. Just dont pay for a consulting company for training as there are lots of dhoti company which provide you free trainings. I heard they provide accommodation too. Good luck!