@msolga,
I don't have much experience with Kaspersky, but I do recommend that you get off your free trial and on to a permanent AV solution, whether that means paying for Kaspersky or another AV product or using a free one like AVG.
I personally tend to use the corporate Norton AV when I can, but honestly as long as you do a few other things (which I will explain) then it really doesn't matter that much.
Ok, now on to whether you should start from scratch again or not. This depends on a couple of things:
1) Do you want to go through the effort of cleaning your computer without formatting? If you have a lot of things on it, and don't mind the trial and error and multiple steps you'll need to take to fix it then by all means do so. But it can also be easier to start from scratch.
2) Do you have a legitimate install disc for Windows? Do you have a disc with your computers drivers? With these two things a fresh install is usually very straightforward.
Anywho, you'll have to make your own decision on that, but I can try to help with either that you decide. Now on to how to keep your computer safe:
1) Do not use your computer with an administrator account. Password protect the admin account and use a separate user account for your day-to-day use.
This one step eliminates almost all security risks. I personally would be comfortable with no AV software, no firewall or anything else but a patched Windows install running a non-admin account.
When you want to install things, you'll have to log in to your password protected admin account, and that can be annoying, but at least you won't have stuff you don't want installed on your computer.
2) Make sure your operating system is set up to receive updates and install them automatically. Unless your software or operating system has a bug that allows people to gain control of your computer, it is pretty darn safe. But bugs in various software are discovered weekly, and several times a year there's a critical one that has the possibility of compromising your computer. If you have automatic updates you are protected from all but zero-day attacks (the ones where the bad guys discover the bug, normally the way it works is the good guys do, fix it and publish the details and months later bad guys use it to get into computers that are not being updated with patches).
3) Make sure you have AV software that is set up to receive automatic updates as well.
4) Don't download and install stuff. No matter how secure your computer is, if you let one piece of malware in it can ruin it all. Remember, a "secure" computer basically just means they can't get their code to run on your computer without you doing it yourself. Thing is, many people do just that, and install this toolbar and that gadget and bundled among that free software is often malware.
Take these few simple steps, and you should not have no security problems on your computer.