Well, if you have a Starbase going by yourself, I highly don't recommend doing any special projects, except for possibly the Featured Projects. Even a small group won't need them, again, except for Featured Projects (if you/they want to do them).
However, to start any project, you will need to go into your fleet window, which can be easily accessed just under your mini-map, it will be a little 'Starfleet' symbol just to the left of the drop down menu arrow.
Click on that, and a large window should appear. On the top of this window, hit the 'Holdings' tab. In the bottom-middle of the page, there will be 5 black boxes, 3 of which will say 'Project Available' on them, 1 will say 'No Upgrades Available', and the last will say, 'Special Projects Available'.
To start a project, click on one of the boxes you see, and to the right, a list of choices should appear, simply go to the one you want, and click a button somewhere on your choice.
To add to a project once started, click on the project as before, and to the right the details for it should pop up. If you can contribute something, the 'Contribute' button should be lit up on each thing needed, if you can give something to it. If not, then you either don't have what it needs or it will say 'Filled', and the button will be blackened out.