Please note that I am not an expert in painting, but rather speak from my own experience and training.

First off, let me remind you that I'm an oil painter. I don't actively use acrylics, and there are people who are much better equipped to explain acrylic paints to beginners. I recommend looking at forums like Wet Canvas to get tips for acrylic paint.

Types of Paint
There are two types of oil paints: water-miscible (also known as water-based, water-soluble, aqua oils, etc) and traditional oil paints. The biggest difference between these paints is their binders, making one easier to clean up with and thin (with water) while the other requires turpentine (or other solvents) to clean up/thin out. You can mix water-miscible paint with traditional, but more than 30% of traditional oils mixed in will make the paint no longer soluble by water.

Brands of Paint
There are many brands of oil paint on the market. My advice is only to find what you're comfortable with and brands that have the pigmentation and lightfastness that you desire in your work.

What I Use
I typically use traditional oil paint, although I have a handful of water-miscible paints as that's what I learned on. Brands I favour include: Williamsburg Oils, Holbein, KAMA, Michael Harding, and the occasional tube from Gamblin. 

For Beginners
When I first learned how to paint with oils, I used water-miscible paint. I loved it for many reasons: clean up was easy, it was easy to learn with, and fumes were minimal due to the lack of solvents required. It allowed me to work in spaces with less ventilation. It won't be for everyone, but it's worth a shot if you don't have a place to work with proper ventilation. It is still used by professional artists, such as Camilla d'Errico. I believe she uses Holbein Aqua Duo (which has some fluorescent pigments I've used).

If you have other questions, please leave them in the comments below.