In addition to what AWK said about IDing by sight, you should make a list of cations and anions. Then take a look at which ones you could mix to get a precipitate and compare to your list of unknowns. You should know the solubilities by your solubility rules. I used to TA a lab like this for undergrads. It's like a big logic puzzle. Here's the idea:
Your list consists of Ba+2 among all the other cations. You know from your solubility rules that if you mix the barium cation with a sulfate, it will give a precipitate. You look through and see that if you were to mix the barium solution with sulfuric acid, copper sulfate or zinc sulfate you'd get BaSO4 as a precipitate. Then you take a look at the other anions. Ask yourself, 'Does barium form an insoluble compound with: nitrate, chloride, hydroxide, thiocyanate or carbonate?" Repeat for each cation/anion pair.
That will get you a long way, but there's a few tricks in here. You have to be careful of redox reactions. Take a look at the reduction and oxidation potentials of each of these that has multiple oxidation states (like copper, etc.). You may very well get a redox reaction that changes oxidation states of your compound.
In addition, you need to be careful with the aluminum nitrate. Not that it's toxic, but it can be a little tricky. The sodium carbonate should tell you if you have an acid, because it bubbles CO2. However, aluminum nitrate may give some bubbles, too, although not like sulfuric acid. This comes from the fact that aluminum hydroxide is slightly soluble, causing the aluminum ion to pull some hydroxide from water, forming a nearly invisible gelatinous aluminum hydroxide precipitate. Of course, you may not note any of this, depending on the quality of water that was used to make the solution, but it's something to keep in mind.