Yes, but my tent came with an extra piece for that purpose. I also have an extra piece of plastic tarp to put down, helps with moisture. But police the site and try to remove any rocks, thorns, rattlesnakes, etc. prior to setting the tent on top of them-helps protect the floor of the tent, and your backside.
most of the tarps are used to keep moisture out. You need to check for sharp stuff before you pitch the tent, but the bottom of the tent should be fairly rip resistant.
You can get really good, MUCH less expensive, clear plastic sheeting at Home Depot and cut it yourself. It weighs a very slight bit more and if you are asking the question, you probably aren't worried about each and every ounce like hard-core packpackers would.
I always put down a thick tarp under my tent every time I camp, just for the extra protection. You don't need anything fancy or expensive. Just pick up a tarp at a number of places. I love REI but no need to spend tons of money.
One tiny bit of advice. Don't put down a tarp or anything that might serve as a "bowl" to catch water under your tent. If you fold edges, fold them under, not over.
Some tents have a tarp fitted for the tent with grommets that the tent poles attach to. These can be quite expensive compared to the "Home Depot" plastic sheet and are usually purchased separately.
For car camping a big blue poly tarp purchased from Academy will work fine for large family size tents. We have also used the clear plastic sheets for inside as the constant walking in and out of tents can abrade the floor and cause it to wear faster.
if you are in your tent a long time, like a cabin tent set up, it probably makes a lot of sense to lay down a tarp (and mind the advice on folding under).
But if you are just going to sleep in the tent, get a sleeping pad for under your sleeping bag. I have more of a moisture problem with condensation from breathing than I ever have from leaks coming from below. Where are you camping?
There is no chance I could fall asleep without a sleeping pad and if you are going to get one, there is nothing I have found better than a Big Agnes. They ARE spendy. The Thermarest pads are fairly expensive also and sound like a good idea, but they are really more for insulation than comfort.