Yes, the English doesn't sound right. "Trains" is usually used as a transitive verb, therefore you could say "my sister is training the swim team." We have an expression, "my sister is in training" if she is preparing herself or being prepared to compete in a sport.
The infinitive is: Тренироваться. Conjugation charts for this and all verbs are on Wiktionary/Vikislovar' - https://en.wiktionary.org/wiki/%D1%82%D1%80%D0%B5%D0%BD%D0%B8%D1%80%D0%BE%D0%B2%D0%B0%D1%82%D1%8C%D1%81%D1%8F (sorry for the unwieldy link).
This conjugation is typical for verbs that have the -ова- / -ева- "suffix" in them - the -ова- / -ева- morphs into -у- (not 100% for verbs spelled that way but it is common).
Technically yes, though in English you don't need the reflexive pronoun because "train" by itself is transitive or intransitive. I wouldn't say it's wrong just not necessary, whereas in the Russian you do need the reflexive suffix because otherwise the verb is exclusively transitive, if that makes sense.