As I'm self-taught I don’t know if I can tell you what the best embroidery fabric is, but I can tell you what I like to use. The main rule - the fabric should not be elastic to not deform the pattern in the process. So here are the fabrics I have tried:

100% Cotton

Cotton is a plain weave fabric. If you want to stitch on a patterned fabric, 100% cotton is a good option. Try to get a heavier feeling cotton if you can (from 145 g/m2 and more). Fabric for quilters are suitable for embroidery.

100% Linen

Linen is a plain weave and evenweave fabric. There are different thread counts for linen. it is a great looking fabric which is great for any project. The beauty of the fabric makes this slight rough surface bearable – this uneven surface is because of the thin and thick threads in its weaving. One disadvantage of linen is that it is expensive. Linen blended with silk is also used for embroidery.

Gabardine

I tried it recently for Paris Eiffel Tower and Cactuses & Succulents projects and I really like it in use. The needle passes through the fabric very easily but fabric a bit delicate and can be damaged by the hoop during your process, so be careful and bind your embroidery hoop. It gives much better fabric tension. You can cover it outside and inside. The fabric is a bit thinner and can tend to bunch a little if you pull too tight, so just be aware of that.

Other

Don’t be afraid to try different things. You can also embroider on jeans, t-shirts (with some iron-on stabilizer), and even canvas shoes! I have successfully embroidered on canvas bags and jeans hats, but my fingers did hurt a little after. There is also batiste, felt, Murano 32, organza, sheer and tulle. If you aren’t sure if it will work, you can always do some test stitches to try it out. Have fun with it!

If you want to experiment with other fabrics you can take a needle with yourself and try some fabrics on your local store, if needle is going through the fabric easily and you feel that it’s going to be comfortable to stitch and it's not stretching - take this fabric for embroidery.

And the last thing I wanted to say - Aida cloth is meant for counted cross-stitch patterns, not for hand embroidery at all!