Learn Finnish: Me tarjoamme kahvia.

Tarjota is a Type 4 verb. To build the stem, you drop the -ta ending and add an extra vowel matching the word's harmony, which is -a here. This creates the stem tarjoa-. Then, simply attach the 'we' ending -mme to get tarjoamme.

Does tarjota undergo consonant gradation like other Type 4 verbs?

Most Type 4 verbs go through reverse consonant gradation, meaning the dictionary form is 'weak' and the conjugated form is 'strong'. But because the core of tarjota doesn't contain a changing k, p, or t, you don't need to worry about consonant gradation for this specific word.

Why is the word for coffee kahvia here instead of just kahvi?

Kahvi (coffee) is an uncountable mass noun. Because we are serving an unspecified amount of it ('some' coffee), it must be in the partitive case. Adding -a to kahvi gives us kahvia.

Can tarjota mean anything else besides 'to serve'?

Yes! Tarjota very commonly means 'to offer'. When talking about food and drinks, it translates perfectly to 'serving' or 'treating' someone to something (such as offering to pay for their coffee).