General rules:
+ 1 Card: Immediately draw a card. This happens instantly, and you have to draw the card before going on. If your deck is empty, shuffle your discard and draw then.
+ 1 Action: Add an additional action to an imagined "Action account" - Do NOT immediately play another Action Card. You need Actions to play Action cards, and every one of your turns starts with exactly one Action on your account. By playing Action Cards, you draw from that account, and add to it every time a card says "+ 1 action" (or + 2 actions etc.). Mark that Action and Action Card are very different things. If you have at least one Action on your account, you may use it to play any Action Card from your hand. Condition: You are not resolving another Action Card right now. Only do one Action card at a time, doing all it says before playing another one.*
+ 1$: You will have one more Coin to spend in your buy phase. Again, this does not require you to do something immediately, but adds to your "Coin account", which you can spend at the turns buy phase and can enhance even further by playing Treasure Cards at the start of your buy phase.
+ Buy: You will have one more buy in your buy phase. This also is only accounted (even though "buy account" probably sounds silly - excuse my bad english) and does not require immediate action. + 1 buy means that in your buy phase you may split up your money to buy one more card, instead of having to spend it on only one card.
So you see, of all +1 X, only + 1 Card actually requires you to do something immediately**. All others cause you to add something to an imagined account that you can use up later on. Actions after resolving the Action Card you are reading, buys and coing in your buy phase.
All other text on an action Card you play has its effect immediately. That is, unless it says so***.
Sorry if i'm trying to be smart too much.
*Some Action Cards, like "Throne Room", "Golem", or "Cultist" tell you to play other action cards. If any card does this, you do not need any action to play the card you were made playing. In fact, playing the second card is part of resolving the first one. See it like this.
** The expansion "Prosperity" introduces another "+ 1"-thing that happens immediately, namely + 1 Victory Token (symbolized by a little crest). But that's it.
***Such effects are mainly in later expansions, but they always make clear when they happen. There are also blue Action - Reaction cards, which are a special thing. They have an Action part above a line that describes what happens when that Card is played during your own turns action phase. This has nothing to do with the part below the line (the Reaction part), and when you play such a card (like Secret Chamber) during your turn, only the part above the line happens. The Reaction part is below the line and always has some "If X happens, you may" statement. This means just what it says, namely that every time X happens, you may do whatever the card says at the reaction part. It does not matter whose turn it is and has no connection with the Action part of a a card, so even if "X happens", you can not do the part above the line.