Basic Dominion has a few things that if you can master them, the additional complexity of Intrigue becomes easier to handle.
First: Three Basic Strategys
1) Big Money: Get Silver when you can, then get gold. When the saturation of money in your deck is 1.6, you can regularly buy Provinces. If you have a province majority, try to remove as many as possible. If you and your opponent are close, start buying duchys when there are about 6-4 provinces left. Do not take the second to last province unless you know they cannot win if they take the last one.
2) Terminal Action Combo: Some cards get you +1 action. No matter how many of these you have, even if you get through your entire deck, you will only be able to play 1 card that does not have +1 action. However, with a card like Labratory, you create an engine that can run through most of your deck in a turn. Buy one terminal action (like witch), and run through your deck each turn, buying whatever is best (usually gold or a province), and you will get your terminal action out almost every turn. One witch goes a long way.
3) Multi-Action Engines: With cards that give +2 action, or duplicate actions etc you can get a deck where you can do many things at once. In basic Dominion Village is a perfect example of this. A Deck with a few villages, a few labs, and some attacks etc can get a lot done on one turn (give them two curses, mine some silvers into gold, and remodel a gold into a province). These decks take a while to set up an efficient engine, but once they get started, they quickly gain enough points to win.
In basic Dominion, just figuring out which of these strategies is best for the 10 cards you have is the first step. Then figure out how the different cards augment or threaten your strategy. Big money can benefit from cards like smithy or council room. Marketplace can add a few crucial extra buys in a competition of action decks etc. Each game turns into a two step contest: BM/Terminal Draw/Multi-Action, and then once you have decided, how to optimize your choice to win.
The next step is realizing that some cards completely change the game.
1) Witch. This slows down every single deck you can be facing. If one player ignores Witches, they will have no chance of winning. There terminal drawing engine will draw into Witches and sputter. The Combo engine even more. Even BM will be slowed enough (hard to get to 1.6 saturation when you have extra non-money cards in your deck). Any game with Witches must include getting Witches as part of the strategy.
2) Gardens/Workshop Combo. This will win before any traditional BM/Engine deck. You workshop gardens, buy workshops, and then try to run out something like estates, which is about the only card you will be able to reliably get. Even witches do not do much to stop this deck, given that every card has the possibility of pushing the gardens to the next point level.
3) Chapel. While commonly seen as the anti-witch, it is actually just a way of beating people very quickly. If you only have 12 cards in your deck (gold and provinces) You will be getting a province every turn. This makes all combo decks run faster, and BM gets to the critical money saturation much faster. If one player uses chapels and the other does not, the game is over.
After this step, of identifying the game-changing cards, you can even go another level. Thief kills Chapel BM decks. You cannot afford to lose one of 4 gold at any point. Chapel does cancel out witches. Etc.
What does this have to do with intrigue? Intrigue just expands the possiblities, but does not add very many additional whole-game strategies.
1) Dukes/Duchies. This can beat a BM Province deck point wise.
2) Torturer is bad enough as a single action, but doubled up becomes horrible.
3) Minion is like witch. If one player is getting minions and the other is not, the minion player will win. it is two free money, and it does not really even take up deck space. You gain the money from as many as you can, use the last to draw 4, and continue on. Having a minion string is awesome.
Simply consider how the new cards fit in with the old. What new combos can you build a multi action deck around (torturer, saboteur, bridge)?
What cards are more effective in a big money deck?
If you had to pick only one or two terminal actions, which ones would they be? What cards help build a chain of actions (Great Hall, Minion, Ironworks sort of).
If you can consider these changes, then you should be fine!