Are there tax benefits to exercising early? Hypothetically say I have some options with a pretty cheap exercise price, and I think the company will be taking on PE money in the near future, AND I have faith that the company will successfully exit in the future, will exercising early save me some taxes?
Yes. If you can exercise at a low price then all of your gains after that will be taxed at capital gains rates instead of ordinary income rates. Note that if you just started you can likely "early exercise" all of your options right now by filing an 83B and paying the company the strike price. You generally have 90 days from the date of the option grant to do this.
The only benefit to exercising early is to avoid the AMT impact.