The first thing you need to do is read. A LOT. Right now, you don't even know what you don't know. Read here, also go to 24fhourcampfire.com, and maybe snipershide. Read everything you can find on shooting and technique. Don't post. You don't know what to ask. You are here saying you are having trouble, yet you ask about buying a good cheap gun. You don't need a new rifle, you need to learn the one you have,and learn to shoot. Don't register on those other sites just yet, especially snipershide, they will chew you up and spit you out.
As for cleaning, get some Wipe-out. It is a good copper cleaner that does not contain ammonia, so it does not smell toxic. It cleans very well, and is easy to use. Give it several minutes to work, and use it until you get no blue on the patch. You don't necessarily need to clean, some barrels like to be clean, some like to be dirty. My long range gun has over two hundred rounds thru it, and is still accurate.
You don't say what you have for base and rings. If you have a weaver one piece base, it will suffice. If you want to stay with inexpensive rings, the Weaver Quad-locks are good. Any quality ring will do you good.
If you bought a package deal, the scope is probably cheap junk. If it is not at least a Banner, get something good. As stated above, the Buckmaster is good. I think it is the best for the money right now. The Bushnell Elite is also good.
When you get a good scope, take off everything and remount . Make sure the base sits flat and true on the action. If it does not, you will have to bed it so there is no distortion. That is as simple as using some JB weld to fill the space under the base. After all the reading you do, you will know what I am talking about. Make sure your rings are low enough that you can get a good, consistent cheek weld on the stock. If you can not get a good sight picture with your cheek on the stock, you will need a spacer on the stock to lift your eye to scope level. You will have to have a consistant shooting style in order to know that it is not you causing the problems.
Use a good set of benchrest bags instead of a bipod. A bipod off a hard bench, it will usually cause bouncing. A good bag set-up, will give you the stability you need.
Factory ammo is kind of a crap shoot. You will have to try different kinds, in order to find something your rifle likes. This may take a while, but it will give you the shooting experience you need. If you find something that shows promise, use it to refine your technique. By the time you get done with all of this, you will have a lot better idea of what you are doing, and a lot of things you do not realize, will be apparent.
If this seems to much trouble, you really don't want to get better at shooting. This is a natural progression with shooting. It really is fun, and the more you do it, the better you will get.
As you go along, keep us updated, and we will give adittional advice and critique. Most here are always interested in helping new shooters.
