Most people don't realize how many paint defects you can actually remove by just using your hands as opposed to any machine polishers. It's easy, safe and extremely cost effective if you're just looking to maintain your own car.
From my personal experience I've found that using Chemical Guys V34 and V36 polishes along with their hex logic hand applicators one can remove some significant scratches, swirling and holograms.
Before polishing or waxing always max sure your work area is as clean as possible. Claying is always recommended if your car hasn't been clayed in a while.
From your description of the defects you'll want to use v34 with an orange hex logic pad. Apply a pea sized amount to the pad and spread it over the scratched areas working with about a 2ft x 2ft section. Start with light pressure and work in circular motions trying to make sure you're working all areas the same amount of time and pressure. If you don't see the scratch giving up continue to increase the pressure until its removed or as invisible as possible. Once the scratch is gone or you've made it as invisible as possible you'll want to slowly decrease the pressure while continuing to work the polish until it becomes completely clear. Wipe clean using little to no pressure with a fresh quality microfiber towel.
Now you'll want to follow this up with v36 and chemical guys black hex logic hand applicator pad. Use the same process as with the v34, except this time you'll just be working out any small imperfections that may have been caused by using v34 (this is technically a compound so it's possibly it can leave behind some imperfections depending on the pressure you had to use). Work in circular motions until polish becomes clear and there are no defects left in the paint. Wipe clean with a DIFFERENT plush and fresh microfiber towel using little to no pressure.
This process will leave your paint with awesome optical clarity that isn't possible with a one step process. Now you'll want to seal in your hard work with a wax or preferably a sealant. Chemical guys offers jet seal 109 which results in awesome protection and adds a great depth to your paint.