Congrats on getting accepted to Yale, that is quite the achievement. You seem focused on Law school and it's great that you have a specific interest already before you even start school. Yale not only is a great school for undergrad, but it also has one of the best law schools in the country. If you like Yale, then I would strongly recommend that you meet with Yale law professors to get to know them and listen to their advice. They are going to know what's best and can serve as great resources for you for certain. The two largest factors of getting into a good law school is your LSAT score and your GPA, those two really make up I'd say 80%-85% of your law school application. While law schools do look at other things, they weigh your GPA and LSAT the most so be sure to do the best you can in those two aspects.
Believe it or not, but "pre-law" students actually do not have the highest percentage acceptance into law schools. Business majors, math majors, science majors, and philosophy majors as a percentage get accepted into law school more often than "pre law" majors. A pre-law related program will definitely get you familiar with the material, but it's not like a mini version of law school. Schools want to see that you are a multifaceted person with interests, they want to see that you are an individual. Think about all the other smart pre-law majors applying to law school who have good GPAs and LSAT score, its a lot. You will stand out to a law school if your pursue things other than pre-law as your undergraduate major. Learn a language, play an instrument, join a club (not the pre law club, join like frisbee ), demonstrate to your future law school that you're not a "one-trick pony".
Interning and gaining professional experience will definitely help your chances so look for opportunities to intern somewhere that will teach you new skills. Interning at a law firm is nice and all, but it is very possible to intern somewhere else and gain relevant professional experience. I interned three times with the U.S. federal government, two of the times were overseas and once was in Washington D.C. for example.
My above comments may seem like I am anti-law school but that isn't the case. The honest truth is that a pre-law major is no better of an assurance of getting into law school than being an English major is. Both my parents are lawyers, both didn't major in pre-law, and both went to law school in the Top Ten.
I hope this helps and I wish you the best of luck!!