What are some database like websites about lower levels of economics (micro/macro) for a research project?
Fama-French library
has good data on which you can test your first Fama-French regression model. 

I have found this video on Youtube that might help you with this first project.

Ps.: It would help a lot if you were more specific about what you want to learn. I can give other suggestions, but I really don't know what you are looking for.