I am trying to mix 1 part water and 1 part liquid margarine on a stovetop. I notice it will not mix too well. I would also like to make this sauce thick. What would be the best way to do that?
Margarine (liquid or otherwise) is mostly plant oils and fats that will not mix, on their own, with water. Even with vigorous mixing you will only get them to stay unseparated temporarily (as with a vinaigrette).
Apart from vinaigrettes you will generally need to use some kind of emulsifying agent to prevent separation. It is difficult to suggest such an agent without a better idea of what kind of sauce you hope to make.
Béchamel sauce, for example, uses roughly equal parts flour and fat to make a roux to which milk is then added. Flour is the emulsifier in this case.
Water and margarine on their own will almost certainly not make a tasty sauce. The closest I can come up with is Beurre blanc which is an emulsion of butter and vinegar or wine. The emulsion relies on compounds that are found naturally in the butter. Although margarine contains emulsifiers, I couldn't say whether they are sufficient to hold a sauce together. Either way, you're not going to get close to Beurre blanc, or anything else with just these two ingredients.
My advice is to find and follow a recipe.
Update: As is so often the case, once I had written this answer I started to doubt myself. Is it really not going to work making a sauce with just margarine and water? Did I really answer the question asked?
Thickened Liquid Margarine (ok, Butter) and Water Sauce
I went to the hob and measured out equal weights of unsalted butter (I've no margarine) and water. I warmed the butter (2 oz) and half the water in a small saucepan. I mixed the rest of the water with a tablespoon of corn starch an whisked this into the contents of the saucepan once the butter had started to melt.
The result is a sauce with the same look and consistency as thick custard. Without salt it tastes very bland, but a quarter teaspoon of salt is enough to make it pleasantly buttery.
Before corn starch was added I saw that the sauce visibly separated after about 1.5 minutes. After it was added the sauce became stable.
Using margarine I guess it won't taste as good, but hey, that's enough guessing.
Chris Steinbach's answer is right. An addition: if you add an emulsifier, you will get an even mix which will even thicken a bit with stirring. You can use lecithine for that. If you want an oil-and-water mix which is not only homogenized but really thicker, use a thickener like xanthan or guar. So, if you relly insist on doing it, there is a way.
But I still can't find a reason why you would want to do this. It is neither easy nor tasty. As the other answer noted, if you want a sauce, get a recipe and follow it. I would start with veloute (beurre blanc is somewhat harder to make, so I wouldn't advise it for a cooking beginner), it is expensive because it uses butter and stock, but the taste is worth it. The method (roux+clear liquid) will even work physically with margarine-based roux and water, but tastewise it won't come near the original.