I've finally fixed the problem with the starting, which turned out to be the battery and alternator. Now I've got what appears to be an oil leak. While driving, the oil pressure drops when turning or standing. I stopped at a gas station and added more oil, then noticed a small pool underneath. When I got back home, I looked underneath the car and observed a slow drip. The car is a 99 Bonneville SE and it appears that the oil is dripping from the right rear of the engine. Thoughts on what part this likely is or where to look when I lift the car?

Sounds like either the rear valve cover or oil pan gasket.

Are those relatively cheap/easy to replace? Would an oil change place be able to replace those?

The valve cover gaskets are easy to do as well as the oil pan shouldnt be to difficult..Im not sure that a lube shop would do these..They are very easy jobs the valve covers are prone to leaking..IF you start at the top of the engine on the back and see oil all the way down it.Then its more than likely gonna be the valve cover..But if just your oil pan is coverd and not the block or heads it would be more toward oil pan gasket

Cool... it looks like the oil'* spilling down the sides of the engine so I'll check out those valve gaskets.

Alright like i said they are easy to replace..Just so you dont have to do it again.When you replace them get the new rubber grommets that go around the valve cover bolts..They do wear out over time otherwise its very simple and straightforward.Take out the 6 or 8 bolts i beleive and pull the old gaskets out then install the new(the valve covers are molded to hold the gasket in place no rtv needed) also make sure you wipe the area down around the valve covers very well first especially above them so no crud falls in when you remove them

The rear one is a little more difficult to get at than the front, but don't let that discourage you. It'* still a very easy gasket to replace. Also, if the rear is leaking the front probably isn't far off. Might not be a bad idea to do both.