The AI in Dawn of War is pretty shocking at the best of times and frequently gets stuck at the first tech level spamming small weak units in an attempt to complete a "tech 1 rush". Typically you will find if you can hold off the initial wave the rest of the game will be a piece of cake seeing as you will have vehicles and they will still only have infantry.
My main point of advice would be to use lots of little squads instead of upgrading one squad (i.e. I would rather have 3 different marine squads of 4 men as apposed to 1 big squad of 10). This means you can engage multiple targets at once and if someone gets drawn into melee when you didn't really want them to you have another couple of squads to help clear the trouble).
As for how to hold off this initial wave it depends on which army you are playing as and which you are fighting. For any army I would definitely say make sure you build a command unit as your first or second attacking unit seeing as the extra HP will definaiely give you a bit more thinking time (this is especially true of necrons seeing as the necron lord is the best unit you have at your disposal) ... from here its a matter of who you are fighting and knowing how they will rush you.
Marines / Choas - Will usually play a solid game with decent AI - harass you a little at the start with marine squads but continue to push through tech levels so you will need to keep up with them technologywise. With choas be careful of cultist squads because they have a tendency to engage multiple squads in close combat at once and leave them prone to fire from another unit.
Elder - they will typically throw guardian squad after guardian squad at you and then throw the occasional farseer whenever they have enough spare rec (strategy = get heavy bolters out asap - or whatever your armies equivalent is).
Orks - Similar to elder, will begin by throwing squad after squad at you but with the added danger of a big mek. This guy can be a pain in the ass because of his teleport ability which will get him to you sooner. (Strategy = you will definitely need a command unit to deal with an ork rush). If orks manage to get to get to tech 2 they are likely to start throwing wartraks at you, counter these with vehicles because their rockets will constantly knock your regular troops over.
Imperial Guard - Again similar to elder, will throw lots of little units at you as well as the command squad which can be a pain because its quickly regenerates powerful units. Heavy bolters (or similar) are the way forward again.
Necrons - On big maps these guys will hardly have the chance to rush you because they are so slow, but you will need to be careful of the necron lord seeing as his teleport power recharges very quick and he can get to your base super fast because of it. Engage this guy with a command unit and have everything else you have shot at him. Probably best to keep his body surrounded as well because when they rebuild him it will revive in the same place it was last killed.
Tau - Watch out for the krut rush, these can be nasty especially if they have feral leap. Lots of separate units will help with this one so that all squads are not simultaneously having to engage one squad.
I didn't ever really play that much Soulstorm so I can't really say more about sisters / dark elder.
It probably is not tilde (~) for you. The tilde on American keyboards is located on the very upper-left corner of the main body of keys (just below the function keys). Try hitting the key above your TAB key, in the left corner, while holding CTRL and SHIFT together. This is the standard "console key" for many games, so keep that one in mind.
I'm honestly not sure why they usually refer to it by the tilde either; on American keyboards, you get a grave character (`) when the key is pressed, and tilde when you hit SHIFT. That would be like saying "Press !" when you really want the user to press "1" on the num row. Ah, well.
Best Answer
Eldar units are fairly fragile to start with, but make up for it with the upgrades available to them and being quite effective in numbers.
As you are talking about early game tactics against an established enemy, you will want to focus on numbers rather than holding out for vehicles and heavier units.
Guardian squads with a Warlock in the unit will increase their damage by 20% and also increase your population cap allowing for more units, helping with what should be your main strategy here.
Your next priority should be upgrades for your guardian/warlock squads. The Warlock can be upgraded with Mysticism at a cost of 150 Req and 75 Pow - This will get them a highly effective close combat weapon, a morale boost to the whole squad and a spell that can immobilise enemies for around 7 seconds.
Dark Reapers are also good early game, again with upgrades making them even more effective. An Exarch in the squad will increase damage by 15%, and fully upgraded optics will increase damage up to 125%.
If you have a Farseer with the Guide spell, it's apparently bugged and can give a squad over 100% accuracy. This coupled with the Dark Reapers ability to move while shooting should be quite effective.
The Optics and Armour upgrades are essential as early as you can get them.
A good resource for stats and tactics on each Eldar squad can be found here.
And some tips against particular armies can be found here.