Am I supposed to kill red guys in Assassin's Creed?

Those highlighted in red aren't necessarily enemies, but guards. In the first Assassin's Creed, you have to be very careful around these guys (it took me ages to learn this). If they are suspicious of something, don't do anything like run, jump or shove - your best bet is just to bow your head and keep your distance. Never run into them, and keep away from them with your horse. So, pretty much, unless you are fighting them or doing a task, try to keep some distance.

Again, they aren't enemies, just very paranoid guards. By default they won't attack on sight, but there is a lot that can make them suspicious.

Templar Knights are also highlighted in red and will attack on sight. Though killing them is optional, there is a finite number of them (30 according to Google) and if you do kill all of them you will get an achievement (at least on the XBOX 360)

You don't have to fight them, but you can if you want. If your objective is to keep a low profile, just walk past them unsuspiciously. However, if you want to play AC like Doom and fight every enemy that enters your field of view, be my guest. The choice is up to you. It's not mandatory, but it is an option.