SJeg studerer (Kan du hjælpe mig?)

We generally don't use active verbs (not in the infinitive form) on their own like this, so I've added a subject.

I dag, har jeg studerert dansk.

Unlike English, we don't add a comma when starting a sentence with an expression of time. You should also use either the present perfect or past tense, since this is something that has already taken place.

Min kæreste hjælper mig med mint dansk, men jeg er stadig ikke god.

"dansk" is a t-word (neuter), so you need to use "mit" instead of "min"

JI dag har jeg læret ordene seng, blad, ske, bord, flaske, skål, og stol.

Because the previous sentence talked about your girlfriend or boyfriend helping you with Danish in general, and not just today, I think it makes it more clear if you add "I dag" to this sentence. Also, if you were going to use the present tense, you would need to put an -r at the end of "lære" (otherwise it's the infinitive form).

Min kæreste viste også mig “det var så lidt”, og det er vigtigt!

"vise" is the infinitive form. But you need to use past tense here. I've also added a comma before "og", because the subject changes (from "kæreste" to "det").