How can you find out if a skin is legendary, epic or something else? Can I distinguish between the qualities by solely looking at the price or are there any other factors that I should note?
The wiki has a good article about skins
As a summary:
There are 7 Prices For Skins (390, 520, 750, 975, 1350, 1820, 3250 RP) and only the 2 most expensive ones have official names: Legendary (1820) and Ultimate (3250).
The community often also uses Semi-Legendary, Ex-Legendary or Epic to describe the 1350 skins since these often were old legendary skins (At the start of league there were only 3 categories: 520, 975 and 1820).
The 975/750 RP skins are often simply called "normal" skins while the 520 or below are often referred to as "minor" skins.
The following is a rule that can be followed but it does not apply for every single skin. Just for most:
A thing to note: Event skins often cost 750 RP. While these are still in the price category "normal", they often do have more special effects than normal or even epic skins.
Then there are also Legacy Skins which used to be normal buyable skins, but now they return once or twice a year.
The rarest "Forever Unavailable" skins
Event-Skins like the Pax or Riot skins (Pax Twisted Fate can cost up to 1000$ nowadays!!)
Reward skins like Black Alistar/Human Ryze (Pre-Order the game in 2009), Triumphant Ryze (Win an official tournament), King Rammus (Play the beta), Victorious Skins (Gold or above in Ranked), Judgement Kayle (Win 10 games in Season 1) and UFO Corki (got automatically rewarded to players in 2009). There Is no way to obtain these skins right now except for human ryze which may be bought on Ebay or similar sites (But expect to pay 300$+)
The "Lost" Skins are often considered the rarest skins. There are only 2: Rusty Blitzcrank who was "Too Ugly" (No joke!) and Urf the Manatee Warwick which was basically an Easter Egg.