This is just one fool's opinion.
I separate the file types into specific folders.
I will have a main Mapping Folder with sub-folders named specifically.
Most of the sub-folders will contain data that will be static, or not edited.
My main folder will contain files that I usually update frequently.
I always have an Archive sub-folder to save files prior to their being changed.
\Mapping
\Archives
\Images
\DEMS
\Shapefiles
\Data_From_Whoever
\Exports
\Mailouts
and so on.
I typically do not rename anything from outside sources unless I edit it.
As for databases, that choice is up to you, and what you are most comfortable working with.
You will find evangelists for SpatiaLite, PostGIS, and so on, and they will all have valid arguments as to why you should choose their preference.
You will probably work with Shape files more than any other vector file since they are the most frequently used type of file (I know, that point is somewhat arguable, but it holds true for me, and many I know.)
Get used to them, and their limitations first.
Then you can experiment with other databases, and see what works best for you.
If you are supplying data to a client, then you want to work with what they desire, make their life as simple as possible. Good clients will appreciate it, and hopefully provide you with more work.
I have a tendency to explicitly name everything that I generate for the first time, then save it with a date appended to the file name when the original file is going to change, or be updated. This practice has saved me quite a few headaches, and has ended disagreements in short order.
The changed files go into my Archive folder.
I also do the same thing when I get data from other sources that has been updated.
Being able to track changes is a very important aspect to GIS.
I know that is redundant, has been said above, but I cannot stress the importance of tracking what has been done, by you, and others enough!
I frequently work on projects that may have hundreds of files, so using explicitly named folders has been the most efficient method for me.
Then, the biggest problem is deciding how much data to put on a map once the file organization is taken care of.
Welcome to QGIS!
I wish you the greatest of fortunes!
