What type of visa should I apply for to enter and exit Thailand multiple times within 60 days?

Just use exemption. Only if you want to enter many times via land border you need a more tailored solution. Then I'd go with the previously suggested 60 day tourist visa with multiple reentry permit. You can do maximum two land border entries on exemption per calendar year.

What is many times??

If it is because you want to visit neighboring countries, then it may be better to go from country to country, without passing Thailand, and then collect the days in Thailand in larger blocs

You could also apply for a 60 days tourist visa and buy multiple re-entry permits at the immigration or airport when you leave. Then you can come and go as you wish within 60 days without any issues.

METV

Perhaps a retirement visa with a multiple entry component. You'll need to be 50yrs or older.

What’s ‘many times’ mean to you? It’s likely you don’t need a visa. Just enter visa exempt

Being from the states you can come and go as often as you like with out a visa until the IO you stand in front of questions why come and go so often.

A multi-entry tourist visa, but you can only get it in your home country.

Depending on what you mean by "many times" and where you plan on leaving and entering, you may not need any visa.

If you're from a country that's eligible for visa exempt you'll receive a 30 day stamp each time you enter. But you can only enter this way at a land border 2 times per year. So if you plan on going across a land border a bunch of times, that won't work.

I'd say as long as it's 4-5 times you're probably fine with visa exempt.

If it will be more than that or at a land border, either get a single entry tourist visa and apply for a multiple re-entry permit in Thailand (less expensive but you might get funny looks from immigration), or apply for a multiple entry tourist visa (more expensive but simpler)

Depends on the purpose