Be careful. Diving into a tower with no minion cover is obviously a bad idea, but running into a group of champions that you don't have an immediate way of killing is just as bad. Once you get better at the game you're going to want to play aggressively enough to keep your enemies off balance as much as possible, but that takes familiarity with the game and the limits of your champion that you don't have yet.
Ideally you should be at least vaguely aware of the capabilities of enemy champions, but since this is difficult when you haven't played most of them at least once yet, play it safe and try to pay attention to what the abilities that they're using appear to do. Most champions have some sort of ability to slow or stun at range, so expect that and don't leave yourself open while your health is low. Try to learn the capabilities of your own abilities, and how to use them effectively against minions and champions.
The in-game suggested items are probably good enough if you're just randomly picking heroes that you have no experience with. I wouldn't worry about developing an actual item build strategy at your point, just grab stuff on the recommended list that looks useful.
Getting the last hit on minions is good, you get gold and more experience than you would have gotten from a minion or tower kill. You probably should be attacking enemy champions as much as possible as long as you can do it safely, but if you can't do it safely (which is most of the time, if the enemies are being careful) you should be trying to deliver as many last blows on minions as possible. Get familiar with the damage you do per attack, and try to make it so that as often as possible, if a minion would be killed by that attack, you target that minion. This takes a bit of practice, but you should be getting plenty of practice on it in the early game. Some champions have area of effect abilities that can deal significant damage to enemies in a large area, these can be very effective against minions if used correctly.
Make sure you understand how tower AI works. Normally, towers will always target minions over champions, meaning that if you're accompanied by a group of minions, you can safely run right up to the tower and attack it directly without needing to worry about it hitting you until the minions are all dead. However, if you attack an enemy champion within the range of a tower, the tower will immediately switch to you. This means that towers are usually safe to stand by if your health is low, though a particularly aggressive opponent or opponents might still try attacking you if they think they can kill you and get out before the tower kills them. This is obviously risky, and I wouldn't suggest trying it yourself with a melee hero unless the enemy's health is low enough that you can one-shot them. If you have a good ranged ability,though, you can probably run within tower range, use it, then run back out without taking much or any damage.
Edit: I'd suggest not starting with any particular hero beyond what the random button or your own caprice provides for you. The best way to understand what an enemy or allied champion is capable of doing is to use that champion yourself. You don't necessarily need to be any good with that champion, just vaguely remember what the champion's four abilities do so that you don't get surprised by anything, like having cannonballs rain down from the sky in a large area (Gangplank) or suddenly going from low health to dead despite having no enemies nearby (Karthus).
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