The four main areas of consideration and adjustment you will have to make are shelter, comfort, cooking and entertainment.
Shelter
Firstly, before you even venture out with your tent, if it's new, you should test it to see if it's rain-worthy. Hose it down before your trip and crawl inside to check for leaks. If you discover any, then you can apply specialist seam sealer to the leaky areas. Although, if your tent is a fairly recent purchase, then there should be no issues with leakage. This is because all modern tents should have a sufficiently high hydrostatic head, making them waterproof even in very heavy rain.
Once you're at the site, check the lay of the land and don't pitch your tent where you can see any evidence of previous water runoff and don't pitch it at the bottom of a slope. Try to find level ground. If it's raining when you arrive and if you have a large tarp with you, fix that up first which will keep you dry when you pitch your tent underneath it. However, don't bother with a tarp if it's at the expense of then having to camp too close to trees or underneath them. Although your instincts might tell you that the tree canopy will provide you with additional protection from the rain, once the rain stops, you'll soon get frustrated by the constant dripping of rainwater from the leaves. More seriously, however, is the possibility of putting yourself and others in danger should the rain escalate into a thunderstorm which could bring down a tree and don't forget that trees attract lightning.
If it is raining when you arrive and a tarp isn't a practical solution, it's best to wait until the rain eases if you have an alternative form of shelter, such as a car and, if you know a storm is on its way, pack up and live to camp another day, unless you have no other alternative.
Rain is often accompanied by wind so make sure that your tent and fly sheet is staked down well and use a natural windbreak when choosing where to pitch your tent to give you added shelter.