If you want to go for engineering colleges out of Maharashtra you can appear for JEE (Main+Advanced) to take admission in IITs/NITs/IIITs and CFTIs. You should have 75% marks in your 12th class in order to be eligible or you should be in top 20 percentile. For SC/ST you should have 65% marks.

There are many  private colleges as well who consider JEE Main score to provide admission.

You can also appear in BITSAT, VIT, SRM, Manipal, Amrita and COMEDK exam to take admission in top private colleges.

To know about the exams in detail you can refer below link

https://engineering.careers360.com/exams

There will be no problem if you change your state after 12th exam.

Good Luck!!