"Gusto kong matuto na ng Tagalog!"  or "Gusto ko nang matuto ng Tagalog!"  (The “na” should go with the verb if it is to mean “already”.  It will not be correct to place it at the end of your sentence because it will be modifying “Tagalog” instead by taking the “that/which is” meaning and making the sentence sound incomplete.  "Tagalog na madali" = Tagalog that is easy.

Ang pangalan ko ay Annicka, at gusto kong matuto ng Tagalog...  – correct, but you don’t really need the comma before “at”

  dahil nandito ako sa Maynila mula 7 years na,  - “mula 7 years na” (since 7 years already) is not correct.  You may either say it as “mula noong 2007 pa” (since 2007 yet) or “may pitong taon na” (for 7 years already).  If you will use “may pitong taon na” it will be better to rephrase this as: “dahil may pitong taon na ako dito sa Maynila”

  at hindi ako marunong Tagalog.  =>  “at hindi pa rin ako marunong mag-Tagalog” (and I still don’t know how to speak Tagalog) or “at hindi pa rin ako marunong ng Tagalog” (and I still don’t know Tagalog).

“Ang pangalan ko ay Annicka at gusto kong matuto ng Tagalog dahil may pitong taon na ako dito sa Maynila at hindi pa rin ako marunong mag-Tagalog.”   (Well, apparently you do know how.  :-) )

Your second paragraph is good.  If you want to replace “fluent” with a Tagalog word, you may say “matatas”.  You may also want to add “nang” (already) before “mag-give up” to mean “to already give up”.  You may use “umayaw” or “sumuko” in place of “mag-give up”.  “Umayaw” means to say “no” or to reject.  “Sumuko” means to surrender.

Gusto kong matuto ng Tagalog !

Ang pangalan ko ay Annicka, at gusto kong matuto ng Tagalog dahil pitong taon na ako naandito sa Maynila, at hindi ako marunong ng Tagalog.

Pilipino ako pero hindi ko kayang magsalita ng Tagalog! Nakakahiya. Nagsimula akong mag-aral ng Tagalog noong September 2013. May konting improvement hanggang ngayon, pero gusto kong maging fluent! Medyo mahirap at minsan, gusto kong sumuko, pero alam kong kaya ko ito.