The Camping page for Yellowstone National Park describes two types of campgrounds: reservation and first-come-first-served. Obviously for the reservation type, you need to make a reservation; at peak times I would expect them to be fully booked. Summer weekends are probably the most popular times of all. For first-come-first-served, you would need to arrive at the campground in the morning, and even then it is possible it will already be full. You would want to have an ironclad backup plan.
Note that people are allowed to stay at a site for multiple days, so arriving on Saturday morning is probably futile, since almost everyone who was there on Friday night will be staying through the weekend.
At a typical campsite in a developed campground, you can expect: a parking spot, a picnic table, a fire pit or barbecue, and a flat space to pitch a tent. Drinking water and toilets should be within walking distance.
In bear country, each campsite will usually have a "bear box", which is a sturdy wooden or metal box with a bear-proof latching mechanism (example). You should use this for all your food, toiletries, and anything with a scent, all the time. Do not store such things in your car; bears can and do break into cars. Popular tourist campsites are usually the most active areas for bears, because that's where there's the most food and the bears know it.
Most people who stay at campgrounds do prepare their own food. National parks generally don't have restaurants within their borders, so you would have to leave the park to get a cooked meal; that might be an hour's drive or more. In some cases, there may be a very small grocery store near the campground; expect limited selection and inflated prices. If you don't want to have cooking gear, you may want to think about meals you could eat cold.
In addition to a tent and a sleeping bag, here are some other things you would probably want:
A foam or inflatable pad to put under your sleeping bag. Otherwise you are sleeping directly on hard and possibly rocky ground, which is not very comfortable.
A tarp or waterproof ground cloth to put under your tent, if it doesn't come with one. Summer rain is common in the US Mountain West, and if the ground is wet or muddy, you don't want moisture to soak through your tent and sleeping bag. Tents are generally not waterproof; at best they will shed rain.
At least one good flashlight. Campgrounds are not lit at night.
Check the weather for the time and place you are going, and make sure your sleeping bag is sufficiently warm. Even in summer, the dry air and high altitude in areas like Yellowstone means that nights can be surprisingly cold. Your sleeping bag probably has a temperature rating; down to the rated temperature, the bag should keep you from hypothermia, but you might still be uncomfortably cold. If you are particularly bothered by being cold, choose a sleeping bag with a rating considerably lower than the temperatures you expect.
A large bottle or jug for water. You don't want to have to walk clear across the campground to the one water tap every time you are thirsty.
Appropriate clothes for protection from sun, rain, cold, and insects.
(Note that this answer addresses "car camping" at developed campgrounds. Backcountry camping, where you carry all your gear and hike several miles to your camp site, is quite different; but it doesn't sound like you are equipped for that, so I haven't addressed it.)
