Learn Portuguese (Portugal): A menina tem uma bicicleta.

This sentence uses the verb ter (to have) to show ownership. If you wanted to rephrase it to mean 'The bicycle is hers', you would use dela at the end: A bicicleta é dela.

What is the root verb for tem?

Tem is the 'he/she/it' form of the verb ter (to have). Since a menina (the girl) is equivalent to 'she', we use tem.

How do I know whether to use um or uma for the bicycle?

Nouns in Portuguese are either masculine or feminine. Bicicleta is a feminine noun (often signaled by the '-a' ending), so you must use the feminine word for 'a', which is uma.

Could I leave off the A at the beginning and just say Menina tem uma bicicleta?

No, because just like in English where we say 'The girl', in Portuguese you need the definite article for 'The'. For a feminine noun like menina, that word is A.