We love to tent camp. Now that we have kids, we primarily do drive up camping. We also do some canoe camping. We used to backpack as well, but it's hard when your kids can't carry much of a load.
Wild animals are an extremely minimal risk. In some areas you have to be careful about where you eat and store your food as it will attract animals. In those cases, you just don't eat in your tent, store your food in your car or hung from a tree.
Honestly, it is people that pose a greater risk. Still, I'd wager that campgrounds are generally safer than hotels. Just don't do stupid stuff like leaving valuables laying around in your tent.
We've camped all over Texas, in Oklahoma, Colorado, Utah, Arizona, New Mexico, Wyoming, Arkansas, Florida, Michigan, Minnesota, Kentucky, Tennessee, Western Canada, and probably a few other places that I'm forgetting. We've never had any serious problems with wild animals. We've had raccoons get into our food, but that's about it.
I would advise you to start small and get into it gradually. Don't make your first trip a month long adventure. Buy good equipment, not
Walmart junk. During your first night out in a thunderstorm with lousy gear, you'll understand why.
Key things to have are a good tent (we love our relatively new Big Agnes), something good between you and the ground (we use thick thermarest pads), and a good sleeping bag if the weather will be cool at night. You should get a pair of sleeping bags that zip together for nights when you want to snuggle. For really cold nights, it's better to sleep separate so that you can go full mummy.
You'll probably also want a good cook set. I like the IsoButane stoves rather than fiddling with white gas. It's a bit more expensive, but it's less of a hassle than fiddling with liquid gas.
Try to establish a pretty good routine for setting up and breaking down camp. If you work well as a team and everyone knows what they are doing, it doesn't take more than 20 minutes to set up or tear down.
On your last day out, try not to back up until at least mid-morning. Early in the morning your tent is often covered with dew. If you put it away wet, it can mildew. Sometimes have you have no choice. When you do put it away wet, set it back up again when you get home and let it air out.
Be wary of cheap public campgrounds near big cities. While you probably envision camping as a cheap way to travel and see nature, many others see camping as a cheap place to go get drunk and place loud music. That doesn't happen often in national parks and nice wilderness areas. It happens all the time in state and county parks near colleges or cities.
Be prepared for rain. A good tent will keep you relatively dry, but the humidity will slowly make everything damp. If you are car camping, you can always hit a laundry to dry stuff.
Some great places to camp that aren't too challenging include Rocky Mountain National Park, Grand Tetons, Yellowstone, Arches, Bryce, Zion, and the Grand Canyon. They all have large camping areas and plenty of support facilities (showers, laundries, restaurants) nearby.
My wife and I have been camping since before we were married. We camped during our honeymoon (including canoeing across a lake and backpacking into the wilderness on the other side). We've camped many times a year every year since then. We've camped in some places just to save money. In other places, we've camped because we preferred it that way. In other places, there was no choice. If you have any specific questions, concerns, or whatever, just ask. Feel free to PM me if you'd like.
Here are a few photos of our camping experiences.
Our tent in Yellowstone National Park. That was in August and it still got down in the low '40s at night.
On the banks of the Buffalo River in Arkansas. It poured on us most of the trip.
This was at Goose Island State Park in Texas. It's a popular winter campsite because it rarely gets cold.