Please note that I am not an expert in painting, but rather speak from my own experience and training.
Types of Brushes
There are many varieties of brushes out there that you can use for painting. Handles come in long and short, there are synthetic and animal hair bristles, and a number of shapes to choose from, including: filbert, round, flat, bright, angle, and fan. There are a couple others I'm not as familiar with that are more typically used in watercolour or gouache painting, or for hyperrealistic detail work.
There are many ways to clean oil brushes. Of course, if you're using water based oil paint and mediums, you can simply use water. For traditional oil paint, soap and water won't work. To avoid using solvents if you lack proper ventilation, use vegetable oil to clean your brushes. Another solvent free option is "the Masters" brush cleaner pads. Finally, there are paint thinners (solvents) or odourless mineral spirits. I have found that using oil to clean my brushes, followed my mineral spirits works quite well, but for certain pigments I will still use turpentine. In any method, the important part is to wipe excess paint from the bristles before attempting to clean them.
What I Use
My favourite brushes are filberts, hands down. I use long handled brushes and while I do own a few flats and rounds, I always gravitate towards filberts.
Find out what kind of brushes you like. Look for both synthetic and animal hair brushes. Until you come up with the type of brush you like, or cleaning method you prefer, don't invest a ton of money. The beginning is where you make mistakes, and no reason to clog a $40 brush with paint because it wasn't cleaned properly. Art supplies will be an investment, but look for student grade materials when you are first learning.
If you have other questions, please leave them in the comments below.