Can I do BE in mechanical engineering and then ME in aeronautical engineering? If not is there any way I can shift to aeronautical field?
Yes. After your b-tech you can go for masters in Aeronautical related branches such as aerodynamics, flight dynamics or power plant (engine) etc.
If you wish to work on aircrafts or helicopter then go for AME licence. It's a maintenance job so professionally AME is a lower profile job when compared with your engineering core job. But in India you don't need to mind about it because people are very excited about live aircraft jobs. This job will pay you good salary only if you take at least 3 licence. Otherwise the salary will be low. The best part of this job is that you will get job very easily in around the world if you cleared particular country's licence requirement.
Next option is that go for implant training/contract job at HAL, NAL, ADA, ADE, etc. So you can work on aircraft related problems. If you want to work on rockets or space related problems then try your luck at ISRO. All of these companies recruite more mechanical engineers than Aeronautical engineers. So check their website for more options. I think now a days they select candidates from GATE pool. So try your best to score good rank in GATE. Best of luck for your Aeronautical career.