The best medium is no medium. If you are happy making your paintings from paint directly out of the tube or having just a thimbleful of stand oil on your palette that is ideal. However if you are using lots of turpentine to thin your paint, that's not good at all. You need to use a medium.
Mediums are used to make the paint flow, control drying and surface gloss and sometimes to level brushstrokes. Here are the three mediums I commonly use.
The first two are alkyd mediums. Alkyd is a paint additive that promotes quick drying and makes for a tough paint film. I use Galkyd most of the time, I thin it by 1/3 with mineral spirits. It seems too thick and dries too fast as it comes out of the bottle. Galkyd is a Gamblin product and I order it from Jerrys' Artarama or ASW. The good thing about Galkyd is that it has a shine, like a varnish medium. The bad thing about Galkyd is that it will dry up in the bottle, so buy a big bottle and decant it into two smaller bottles and fill the one you are storing all the way to the top of the neck including as little air as you can.
Liquin made by Winsor & Newton is the other well known Alkyd medium. It handles well right out of the bottle but you can thin it to make it last longer. It is a thixotropic gel, that is, it is a gel till you push it with your brush, then it liquefies, when you take your brush away it gels again. Liquin has a somewhat matt finish, so it needs to be varnished to look as good later as when you painted it, and some people find it irritates their nasal membranes.
One thixotropic gel is Marogers medium. I do not recommend you use Marogers, there is a group of artists from New York that use the stuff. Jaques Maroger was a curator of painting at the Louvre in about 1900 who believed he had found the secret of the old masters , He cooked what is called litharge and black oil. It involoves heating lead, and is extremely poisonous, I have no problem painting with lead. I do have a problem with cooking it. Marogers is by most accounts impermanent, although it does impart a beautiful transparent body to the paint, and gives a silky sort of handling. Gamblin makes a substitute. I believe they call it neo-meglip. Meglip is another name for the sort of gel that Maroger popularized. Marogers users have a cult like devotion to the stuff and should never be confronted about its use. There are some painters out there doing great work with the stuff though. If you are already using Marogers and wish to continue, hey, I warned you.
While I think alkyd is great medium and use it most of the time it is the product of modern chemistry and is not a traditional medium.
The most common traditional medium is usually referred to as VTO. It is a mixture of equal parts damar varnish and stand oil (sometimes linseed oil) and 3 to 7 parts TURPENTINE. I stress turpentine because mineral spirits is not a good thinner for damar varnish, I f you use damar in your process, your thinner must be turpentine. Every artist who uses this medium seems to tweak it to their own liking. If you use too much oil it can give you a surface that is sticky like flypaper.
I have used turpentine from the hardware store for years, however finding quality turpentine has become difficult. Much of it no longer smells sweetly of pine, but has a dreadful odor. It still says gum turpentine on the outside but I don't know if it really is. There are still some brands that are right. I take it to the counter and say something like "does this turpentine smell like death?" They will usually let me open the can, (including its little inner metal hymen) and smell it, if it doesn't smell like pine I let them smell it and refuse to buy it. If the clerk is a totally uninformed and unconcerned high school kid I can't pull this, so I try to deal with the manager or the old guy who runs the paint department.
There are times when you want your paint to stay open for a long time (dry slowly) poppy oil works nicely for that.
Years ago I used Taubes copal medium. Then one day I went to the art store and bought some that was totally different I quit buying it. The quality of most of these materials is constantly changing and you have to watch out for that. If you want that enamel like copal finish you can still get copal tears from Pearl paint, along with most other exotic vanishes like mastic etc. that I really don't have a use for, but you may.
Most of the time I use deodorized mineral spirits from the hardware store. Gamblin makes a nice one called Gamsol. Double rectified, English or artists grade turps can be had from online suppliers and is a really fine product. As oil paint is phased out in house painting we may have to buy this when we use turpentine. It is of course much more expensive.
I guess a word on permanence is required here. I have known plenty of artists, who grind their own paint and use only copal tears cut with turpentine made by Benedictine monks in their underwear. These artists make paintings which will last forever but should never been made in the first place. Its like some guys I knew in the 70s who had these incredibly expensive stereos , but only had six albums, two of which were by Kansas. They were so hung up on the technology that they forgot it was about the music. I had a very ordinary stereo but I had a LOT of music. The point I am making here is that, it is about the pictures more than the process. You can make yourself really crazy over materials, and plenty of folks do. My guess is that alkyd paintings will outlast just about anything else anyway. My paintings are bulletproof. I will return to the subject of materials again as I have addressed only the basics here. I hope though I have told you enough for you to have a general understanding. I am trying not to be overly technical this early in the blog.