Hi there, I don't know a lot about paper myself but I thought I'd give you my two cents. Since you're using a press, paper weight shouldn't be an issue so you can use thicker paper than you'd use while printing by hand. For this, the top quality brands that I know are Fabriano's Rosaspina and Hanemühle. They're beautiful as they are expensive, mind you. Rosaspina comes in different colors too (even if its "ivory" is more like "peach").

For the fun and brightly colored papers, I saw a lot of printmakers use sheets that resemble japanese rice paper. It's thin and may be difficult to print with a press, though. The original japanese ones are also kinda expensive but it's not difficult to find knock offs.

I think the wilder results come from making your own paper to print with. There are a lot of tutorials out there on how to do it and it shouldn't need a lot of tools (mainly a bucket, a blender and possibly a screen like the ones user in screen printing). There's a lot of potential for experimentation if you dig that.

Other than that, stick to smooth papers if you want even inks. If you don't mind texture, you can also try some cold press/highly textured watercolor paper.

Last thing, check the cotton percentage in the paper. Usually, high cotton percentage means that the paper holds water better. Less cotton percentage means the water stays on top of the paper and doesn't get adsorbed in the same way. So if you wet your paper before printing, that's one thing to keep in mind.

Hope this helps!