Just so you know, beating the standard computer is basically nothing . . . you probably want to get to the point where you can beat "hard" or "harder" computers before going online if you don't want to get your ass kicked a lot.
You should really get the grid hotkeys. They'll help your micro immensely since the original setup would have you moving your hand all over the keyboard.
Here's the one for original DoW
http://www.cokane.com/warhammer/gridhotkeys/KEYDEFAULTS.LUADownload this file (right click and choose save link as . . . ) and put it in dawn of war/profiles/profile1/w40k
and for Winter Assault:
http://dow.lerp.com/gridwxp7.zipJust extract this one to your dawn of war folder.
This will reassign your hotkeys so that the buttons correnspond with the left side of your keyboard, like this:

With that out of the way, I can give you some general tips.
First of all, do yourself a favor and don't play any game that has "quick-start" enabled, because this will make everyone start with max resources, effectively destroying much of the strategy and balance of the game. A lot of n00bs like to play this way, for the reasons I just mentioned.
As you should know by now, the resource system in DoW is directly tied to map control. This means that the game is very oriented toward aggressive play and turtling is never a good idea. You should always be fighting your opponent for control of the strategic points, don't just sit because the longer you do so, the more he will gain an advantage on you in resources.
Listening posts are very important in that they both prevent an enemy from capturing your points, and give you added resources. Many new players don't know that upgrading your listening posts not only equips them with weapons, but also increases the rate of requisition you get from them.
You should also know about
overwatch. This is a feature that goes unnoticed by a lot of people. Basically, when you right click a production or reinforcement button, it will begin to flash white, and will then constantly produce that item as long as you have the required resources. Here's an example:

This trick can also work for producing new squads or heroes. For example, if you don't have enough resources for a hero/unit, but want to get one soon, you can simply right click on the portrait and it will automatically be built once the resources for it become available.
Those are some general tips/tricks for you to get started, to go into more detail tell me which race you want to learn about, because they differ in a lot of ways and it's difficult to come up with general strats that work for all of them.