Can I install tandem breakers to stop my breakers from tripping?
Whether you can install tandems depends on the panel. Some panels are not listed for their use. If you look at the panel label and see a number that ends with 3040 it could use tandems. If it is like 3030 you could not. If you post the panel model and number someone could give you a more definitive answer.
Yes, a tandem is 2 breaker internals inside of one case. Your assumptions are correct.