My Grandfather currently uses a Windows phone (bought used online- came from Italy) on FamilyMobile which uses Tmobile service. His internet has never been great. Is it because I need a phone that specifically works with Tmobile frequencies?
If it came from Europe then it most likely is missing some of the T-Mobile frequencies.
You can check on willmyphonework.net which bands that phone has.
Besides that, since it's only on a T-Mobile MVNO the data is probably slowed down, and even though it runs on T-Mobiles towers, it still has much slower data then a regular postpaid T-Mobile account.