lens questions!

hello! i am currently saving up for the a7iii. i have a few lenses in mind, but im having trouble understanding some of the abbreviations/mount types. i take pet portraits/action photos, so i really do want to save for the 135mm.

my starter lens will be the 50mm, but i see multiple types. one says "sony a fit" and the other says "fe." will this make a difference? is the FE compatible with the a7iii?

also if you have any lens recommendations that would be great!! thank you!