I´m a learner too, but I think it should be like this:
1. Mehmet´le geçen hafta tanýþtým - Seems perfectly ok to me!
2. Only a small correction in second sentence: Onunla geçen cumartesi tanýþtým.
With me would be "benimle", with you (singular) "seninle", with us "bizimle", with you (plural) "sizinle", with them "onlarla". And you don´t have to put apostrophe after this word, it is only used when appending suffixes to names, like in: Mehmet´le (as you have written), Mustafa´nýn (Mustafa´s), Türkiye´de (in Turkey). But if word is not a name then you simply append suffix to word: bizimle, onun, eve ...
