Can I travel within the US with an expired visa?

A U.S. visa (visa stamp) is only for entering the U.S. The visa's expiration date is the last day you can use the visa to enter the U.S. The visa has nothing to do with your ability to stay in the U.S. That is determined by your status, which is given upon entry. Not having a valid visa does not affect your ability to stay in the U.S., as long as you have a valid status; and conversely, having a valid visa does not mean you can stay in the U.S., if you are out of status.

The status and length of stay are determined by the I-94 you are given upon entry (well, you are no longer given a paper one anymore; you can print it out online). F-1 students on multi-year programs are always admitted for "D/S" (Duration of Status) on their I-94, meaning they do not have a fixed length of stay. Rather, they remain in F-1 status as long as they have a valid I-20 and do not violate the conditions of F-1 status. (Similarly for J-1 status, they remain in J-1 status as long as they have a valid DS-2019.)

You would only need a visa if you leave the U.S. and need to re-enter the U.S. again. You an only get a visa at a U.S. consulate outside the U.S. It is common for long-term international students in valid status to have their visa expired for years because they have not had to leave the U.S. during that time.

When traveling within the U.S., you should almost never run into anyone who will ask about your immigration status, unless you go within 100 miles of the Mexican border where there are Border Patrol checkpoints. Local law enforcement (except perhaps in Arizona) generally don't deal with immigration stuff because it is a federal matter. It is not within the TSA (the agency you will most likely deal with when traveling by air)'s jurisdiction to check immigration status, although there have been occasional reports of bad agents who yell at people for expired visas (not understanding that the visa has nothing to do with status). (Plus they do not have the training to check even if they wanted; for example, someone out of status is still legal if they have a pending Adjustment of Status, Change of Status, or Extension of Status application, or if they have pending asylum or Deferred Action, etc. The intricacies of immigration law are way too complicated for a non-immigration agency to deal with.)

Your I-94 and I-20 is sufficient proof that you are in valid status, if you ever need to prove it. You should bring that with you.

(shorter version that might actually get read)

I am an international student and authorized to stay in US for 2 years on my I-20 form.

2 years from when? If you arrived after 2013/1/1 you're fine.

My visa expired on June this year

Your visa, that you showed to immigration on arrival? Or your I-20?

Your visa expired in june, it was used when you entered the USA (and thus cannot be used again).

All you need for domestic travel is valid ID, but unless you have a local driver's license you will most likely use your passport. All the airline and security will care about is the name matches the ticket and your face matches the picture. Enjoy the trip.