The first one that is the small station is called J1772, and yes, no problem charging with that. I think the cars still come with the adapter for that. It's going to be a bit slow, like maybe 20 miles per hour.
Thanks for the feedback. If you mean 20 miles of driving for each hour of charging, that is not a bit slow, that is glacial.

It is also a little disappointing. Moms is one place I go regularly, so if it was effective charging, that would have been a nice fit. Sounds like that is not the case.
The second big station, I can't quite see what kind of handles those are. There are two types for those DC fast charging stations. One is CCS, and Tesla doesn't have any kind of adapter. But the other is CHAdeMO, and Tesla does sell an adapter for $450 that can use that.
CHAdeMO Adapter
A lot of those stations with the two cords like that are one of each handle type, so that looks like you could use the CHAdeMO side if you want to go for that adapter.