Pardon my newbie question, but can Tesla cars use other charging stations with the adapter that comes with it?
It depends on the type of charging station you're talking about. If it's a Level 2 (L2) charging station, then almost certainly yes. The vast, vast, vast majority (99.99%) of those in North America use a connector called a J1772 plug. Tesla provides a J1772-to-Tesla adapter with each of their cars sold so that all those stations are usable. For L2 charging where there isn't any hardware and only an available electrical outlet, the answer is mostly no. Tesla sells plug adapters for their Mobile Connector which work on the most commonly found outlet types, but, except for the normal US wall outlet (NEMA 5-15) adapter, these all have to be bought separately.
For Level 3 (L3) or DCFC (Direct Current Fast Charger) stations, the answer is always no. Teslas don't come with the adapters. L3 chargers in North America that aren't Tesla Superchargers use the CHAdeMO plug and/or the CCS1 Combo plug. As others have already replied up thread, Tesla sells a CHAdeMO-to-Tesla adapter that works for all current Tesla models. But they don't currently offer a CCS1 Combo-to-Tesla adapter at all, though this is thought to be in development.
(note: Tesla does have/sell an adapter for CCS Combo-to-Tesla for the European market which uses the CCS2 Combo standard)