I have been to Italy for 2 months since January last year.
We have been in Italy for 2 months since January last year.
--> Well, in my opinion, there's a problem with time here. I mean, "since January last year" means at least a year (=12 months) has passed. So "for 2 months since January last year" seems illogical.
I prefer to start to studying Medicine.
She prefers to starts studying Medicine.
--> I would simply say "I prefer studying Medicine./She prefers studying Medicine."
I studied laws the last year. / the last weekend
--> I studied laws last year. (without "the")
They didn't participate in the tennis match recently.
--> I would use Present Perfect Tense instead of Simple Past Tense, i.e "They haven't participated in the tennis match recently."