Ha, I once dated a guy where I did most of the travel, in my case a commuter train. He didn't drive either and I did. I put up w/it for a while as my family lived in the area, but I also got tired of it after a while and broke up with him. THEN he started coming to visit me. I even got a proposal, LOL. It was all very icky.
Anyway OP, no, its not his job to chauffeur you around. Expecting him to do so is inconsiderate, which is unlady-like. That's a lot of driving for your BF, esp if he is taking you home as well. 3 hours of travel for him by my math. 45 min each way X 2. Plus the fuel costs as mentioned.
So,
1. Gas is expensive. Offer to pay for some.
2. Take the bus there, esp if it is daytime/safe to do so. Mbe he can take you home if the cost is about the same. That's still 1.5 hours of travel for him (take you home and go back home himself)
3. Get a BF who lives closer to you.
What your gram is suggesting is you find a guy who is a pushover. This won't work for either of you in the end.
Second thoughts can generally be amended with judicious action; injudicious actions can seldom be recovered with second thoughts.
--Cyteen by C.J.Cherryh