Congratulations on becoming a US citizen!
In Texas, you do not have to choose when you register to vote. In other states, you must. When you vote in the primaries in Texas, that is when you are considered Democrat or Rebulican (or whatever). So, for example, if you choose to vote in the Democratic primaries, then any other primary election this year, you would only be able to vote on the Democratic side.
I'm not sure if you know this, but in the Primary elections, the folks running against each other are all in the same party. And the primaries are designed to choose the top guy (or gal) for that party.
Then in the general election, the candidates are running against the top guy/gal from each of the other parties. So there is only one person from each party on the ballot. At this point, regardless of what party you chose to vote for in the primaries, you can vote for any candidate on the ballot.
Hope this helps. Congratulations on wanting to get involved.