Learn Finnish: Tää on mulle tosi tärkeä. - This is really important to me.

Tää is the common spoken language (puhekieli) version of tämä (this). In everyday conversation, you will almost always hear tää instead of tämä.

Is mulle the spoken version of minulle?

Yes! It is the allative case (meaning to me or for me) of the spoken pronoun mä (I). In formal written language (kirjakieli), this would be minulle. The allative case is used here with tärkeä to express who the thing is important to.

I usually see todella translated as really. Is tosi also a spoken language feature?

Exactly. While todella is standard and works in all situations, tosi is extremely common in spoken Finnish as an intensifier meaning really or very.

Why is tärkeä in the basic nominative form rather than the partitive tärkeää?

Because the subject tää (this) refers to a specific, countable thing here. If you were making a general statement about an abstract or uncountable concept, like sleeping is really important (nukkuminen on tosi tärkeää), you would use the partitive.