Problems creating correct file format for Squarespace business logo

Hi, I’m fairly new to Illustrator and I’m just using it to create my business logo. I’ve done this and I'm happy with the result. Just having trouble with exporting.

In particular I’m having an issue with my Squarespace site.  I would have thought that a vector file would be perfect for a website logo however Squarespace only accepts the following: jpg, png or gif.

I’ve created a large png file - much larger than the size it is displayed on the site, but its still looking a bit crappy around the edges… subtle but noticable.

Here are the file details (from "get info"): 

Kind: png

Size: 168kb

Dimensions: 3123 x 3123

Colour Space: RGB

Alpha Channel: Yes

Any ideas?

Also:

It seems that I should have a few different versions exported in a folder ready to use for different uses: business cards, business website, social media, other places on the web etc.

What would be an ideal little kit of export formats & sizes?

I’m using illustrator CS CS6 v16.

Thanks!