I am boiling the water, and then adding my spaghetti into it, and cooking for 7 - 8 minutes. It all works fine. However since am new to cooking, I don't know how to use the store-made tomato paste. Do I just add it to the pasta after it's done? Or do I add it while the pasta is being cooked for 7 - 8 minutes? Or do I cook the store made tomato sauce by itself in water in a different pot?
Oh, that is not a real tomato sauce, this is mostly a tomato mush with a bit of salt. (The term "tomato paste" on the jar might a bit misleading, this product looks somewhat thinner.)
In a pinch, you can use it similar to a store-bought sauce by heating it gently in a separate pot on the stove (or in a microwave) and serving it with your spaghetti. But for a real sauce, it is probably too bland, missing spices and too acidic. Not what one expects from a sauce.
If you are really inexperienced, do a quick websearch for "easy tomato sauce" or similar. Many use tomato paste and you'll at least get a quick idea on how to use it, together with herbs and spices. In short, you want probably a bit of garlic, some herbs like basil (or even "italian seasoning"), possibly salt and pepper and - depending on the consistency of the tomato paste - a bit water (scoop a bit out of your pasta pot). If it's too acidic, add a pinch of sugar, not to make it sweet, just to balance out the acidity. Either toss the drained pasta in the sauce to coat or serve the sauce on top of the noodles.
It is real, as tomato paste is a concentrated form of tomato puree. Italian red/tomato sauce is often made with a combination of tomatoes, tomato sauce, and tomato paste, none of which are seasoned at the start. The ingredients are simmered down, starting with a bit of olive oil in the pot, and with seasonings added (salt, pepper, oregano, marjoram, thyme, basil, rosemary). Many cooks taste the sauce to check on acidity, and add a bit of sugar to adjust it. Garlic and red pepper flakes are optional (and often depend on one's Italian region of origin).
Use what you have, simmering it with the dried seasonings that you have available. After draining your pasta, return it to the pot and add the sauce, tossing it over heat to allow the pasta and sauce to incorporate (and thinning a bit with some reserved pasta water, if needed). Off heat, add cheese and serve.