There used to be lots of ways to register in Texas, for example when you renew your driver's license you could check a box to register yourself at the address of record. Mail in applications, walk in at county courthouse locations, and there are sometimes mailings from interested non-partisan groups that include the forms. I don't know if any of those are still allowed. The trouble these days is that the GOP is trying to remove registered voters by purging the rolls if people haven't voted recently or they suspect someone isn't eligible (for whatever cooked-up reason.) Then you have to be sure to get yourself added back in. So yes, checking to be sure you're still registered is important.

It varies from state to state, the more liberal locations make it as easy to register, and also to vote. Mail in, for the big one. During COVID lots of places had drive-through voting and in Texas the Secy of State fought that (because it was done in the Democratic region of Harris County, where Houston is situated).

I can check online to be sure that my mail in ballot was accepted (you put the ballot inside a larger envelope on which your voter ID and signature are there for confirmation that you are voting in your own name.) If it is rejected for some reason you can go to the polling place on election day and vote. But you have to know that it was rejected in order to do that.

Some places have same-day registration, so if you're new in town you register and vote at the same place. Not here. In Texas you have to have registered 30 days or more before the election.