If you are talking about swirls rather than deep scratches then you have two options. One is to fill them and the other is to remove them. If you fill them then the fix is temporary but still effective for the short term. If you want to remove them completely then that is a lot more labour/cost intensive. I bought a DA polisher a year or two ago and it has made things much easier but your cost shoots up and your product range expands significantly.. I wouldn't recommend trying to de-swirl a car by hand, unless you have a lot of time and don't mind a bit of muscle pain (I've tried it!!)
If you want to fill swirls then I'd go with autoglym super resin polish as it contains a fair bit of fillers and is readily available and not too expensive. It needs to be worked into the paint as you need to break the fillers down into the swirls, so its not just a wipe on wipe off if you want to get the best out of it. Then a coat or two of their extra gloss protection to seal the finish and keep the fillers in place.
My favourite polish at the minute is Menzerna 203s. It has a reasonable amount of cut and also finishes down quite nicely but I've only ever used it by machine. Had quite a bit of success with VW and Merc paints, not tried it on my Audi yet, maybe when it warms up a bit!
If it's deep scratches you can feel with your finger nail then I would just try and improve the look of them rather than remove them. If you were to magnify a scratch the paint would have rough edges along the scratch, these catch the light which make it more visible. If you can smooth these out (and probably clean out the scratch at the same time) it will be much harder to see, assuming you're not down to base coat/primer/metal etc. For this I'd try Scratch X or a similar product rubbing along the scratch, you don't need much product!
The important thing is to not to try and nuke the paint! Go with the weakest combination of polish first as you may be surprised, if that doesn't work try again then go up a level. Also when you are polishing think of the polish as grades of sandpaper. If you went at your dining room table with 60 grade it would look pretty rough, you'd then have to follow up with finer grades to get the finish back. Also after a while the sandapaper loses it's effectiveness as it wears down. (If it's not already clear I am using an analogy, I am not saying use sandpaper on the car lol) Same with polishes, they need to be worked in order to do their job, you need to use them until the abrasives have broken down, normally when they have gone clear. Then you can wipe them off. A long old job but worth it in the end.
Never attempted removing bird stains.
