Learn Swedish: Jag ska besöka henne på sjukhuset. - I will visit her at the hospital.

In Swedish, we generally use på (on/at) for public institutions, workplaces, and services like sjukhus (hospital), bank (bank), or biograf (cinema). If you said i sjukhuset, it would sound like you are specifically emphasizing being physically enclosed inside the building's walls, rather than just being there as a visitor.

What is the -et at the end of sjukhuset doing?

That is the definite article, meaning "the". In Swedish, "the" is attached to the end of the noun. Because the word for hospital is an ett-word (ett sjukhus), it takes the ending -et to become sjukhuset (the hospital).

Why isn't the verb besöka (visit) in the present tense?

It follows the helper verb ska (will/going to). Just like in English where we say "will visit" and not "will visits", the main verb after ska must always be in its base dictionary form (the infinitive), which is besöka.

Is there an easy way to remember the word sjukhus?

Yes! Swedish loves combining simple words to make new ones. Sjukhus is made by combining sjuk (sick) and hus (house). So a hospital is literally a "sick-house".