On an older vehicle that’s hasn’t been detailed in a long time, it will be a multi step process. First wash then clay, after that you will probably need to use compound, then polish then wax. Think of the steps as sanding the clear coat to a smooth finish as you would a piece of wood. The compound will take out some of the deeper scratches but if you can feel it with your finger tip you will not be able to get those out. Next use the polish to remove swirls and spider webs. Lastly, wax the vehicle.
Additionally there are synthetic clay bars that are more sponge like and are easier to use than the actual clay and wont be ruined if dropped on ground. Just make sure you use plenty of lubricant such as detailer and you wont have a problem with the clay.
I have done numerous vehicles but use a dual action buffer and pads. Harbor Freight sells an inexpensive one made by Chicago. They also sell buffing pads. There is one pad for each of the steps of compound, polish and wax. You will be amazed how well the vehicle will turn out following these steps. It can be time consuming but the results are extremely satisfying.
After this process you may realize the price the local detailer is relatively cheap.
If you plan on doing vehicles several times a year, it can be a money saver doing it yourself. But the up front cost of the buffer, pads, clay bar, detailer, compound, polish and wax is expensive, more than the cost you were quoted.