Bullett bizim çok güzel Jack Russell cinsi köpeğimizdi.
( I used translate for köpeğimizdi.)
Well, you were supposed to tell a memory about him/her :) I suppose you are dipping your toe in the water, if so this is a good start but try to write longer. Don't be afraid of making errors, everybody does. We (as native Turkish speakers) correct them gladly. You can use translators btw, though they are not very good with Turkish.
The sentence is not wrong but not native-like. I think a native would say something like: (Ben çocukken / Eskiden / 10 yıl önce) Bullett adında bir köpeğimz vardı. Çok güzel, Jack Russell cinsi bir köpekti. (S/he died or is not with us anymore as the sentence is in past tense). English speakers tend to use too much pronouns. Try to avoid them when they are not necessary (when one get the meaning from the suffix or the context).
Btw I am very interested in learning about South Africa and about why you want to learn Turkish. Maybe you write about these subjects, both in English and Turkish. Feel free to ask any questions. Good luck.