Yes, you should have cycled your tank first. Usually takes 2 to 4 weeks before bacteria builds up to help control nitrites, etc. But, since you have fish in it already, the best think to do is wait a week or so and see what happens with your water quality. How long has it been running now?
The cobweb-like strings could be several things, including algae. I'm going to assume that you're using an urban water supply, complete with chlorine and other chemicals. Did you add a chlorine remover or other conditioner?
As for the fish, I doubt anyone can identify them with out a pic, but there are numerous sources for pics online, including this site. There are more types of tetras than I can name though, so who knows.
Either way, it seems the seller was horribly remiss when you bought the tank and fish. He should've given you much more info. Trying to make a quick sale, I guess.
All you need to clean your tank is a gravel vacuum. Vac the gravel and change 1/3 of the water every 2 weeks or so, after it becomes established; more often if you add more fish. Wash your filter media in your dirty tank water when you do a water change. Don't wash in clean water as that will deplete any good bacteria living in it. A conditioner with every water change shouldn't be necessary once the tank is established, but won't hurt anything either.
If you do indeed have tetras, they usually prefer a larger group; 5 at least. If you have platies, and have both sexes, or if you have an already pregnant female, your tank will be full of fry soon enough.
3g betta, 29g molly, 90g Amer cichlid, 90g community, 137g African cichlid, 220g water garden, 6,500g koi pond