I'm not new to writing, but I'm new to writing for furries. What terminology do I use? Do I use special grammar or words like in an MLP Fic? Do I name the characters differently or do I name them with normal human names like, John? Any advice you can give me?

*rubs hoofhands in glee "Now we're talking!"
Some of the terms are setting-specific; some use "Furs, furson, fursonality, everyfur" others use "Furkin" (especially for the "Furkindred" setting), and sometimes "Morphs" (for Anthropomorphs)
If you want to be PC from the non-mammal viewpoint, reptiles are "Scalies", birds and bats are "Avians", and finned friends are "Aquatics"
Equine furs such as my avatar often have broad hoof-like nails called hooftips instead of fingertips.
Anthros with more animal like digitigrade legs are sometimes called "jack-legs" (humans have plantigrade legs)
(example of digi legs; )
"Are you kidding?" may come out "Are you pulling my tail?"
And so on... If you have anything specific come to mind, feel free to ask.

Oh yeah, on names, it depends on the setting. Anthro Equestria might get a mix of normal human names and typical MLP, like "Scott Skybright". Sci-Fi settings have had slightly eccentric spellings like "Jozeph", "Mauri" "Miki" or standard modern names. There is nothing wrong with using typical modern names in most furry stories, btw.