You're just getting into metal, so I think it's best to stick the more mainstream bands for now. Listen to compilations of classic metal songs, try not to dive straight into blackened death metal, slam or funeral doom. I'll try listing subgenres in order of accessibility.
If you want clean vocals and simple song arrangements, I suggest
Heavy Metal,
Power Metal,
Gothic Metal and
Alternative Metal. Bands like Iron Maiden, Blind Guardian, Nightwish, Faith No More and so on. Some
Doom Metal like Candlemass falls into this category too, but it's an incredibly diverse genre. Before you know it there's bands with 40 minute long songs filled with guitar feedback and shrieking, as well as riffs that last an eternity.
If you like the clean vocals but want some more complex songs, try
Progressive Metal. Ayreon showcases many different metal vocalists in a wonderfully cheesy rock opera package. Dream Theater and Symphony X are well-respected bands with
insane amounts of talent, and they use clean vocals exclusively. If you want a taste of death metal-style songwriting while still having clean vocals, check out Nevermore (Jeff Loomis is one of the best guitarists in metal, in my opinion).
If you want to delve into more extreme vocal styles while still not plunging into the deep end, check out
Thrash Metal and maybe
Melodic Death Metal. There's the big four of thrash (Metallica, Slayer, Megadeth, Anthrax) but there's also bands like Pantera and Sepultura who also pioneered
Groove metal (essentially thrash with the tempo lowered). As far as melodeath goes, you can't go wrong with In Flames (earlier stuff), Soilwork (latest couple of albums especially), Dark Tranquillity and At the Gates.
Ready for some more intense stuff? Check out
Death Metal and
Black Metal. Black metal bands like Dimmu Borgir and Dissection are intensely melodic and are fairly accessible (as are Emperor). If you want more extreme black metal without dealing with the poor production values normally associated with it, check out Der Weg einer Freiheit.
Folk Metal usually has a black metal influence, some good bands to get into are Ensiferum, Xanthochroid, Equilibrium and if you're feeling particularly ambitious, my favourite folk metal band, Moonsorrow (get ready for half-hour long songs though). As for death metal, can't go wrong with early 90s Carcass, 90s Death and Amon Amarth if you want something more fun. My favourite death metal band is Cryptopsy but they're not for the faint of heart. Also check out Opeth if you want to hear how far the death metal sound can be pushed.
There's a lot more to explore after that, but I'd rather not indulge too much. Check out compilations, search for the bands that helped define these genres in the first place. If you don't like how something sounds, maybe come back to it a few months down the line. Metal can be an extremely acquired taste.
President Bagel said:
Avoid metalcore, deathcore, crabcore, noisecore, emocore, and piratecore like the plague. They are all impure and false. Some grindcore is pretty good, but I find most of it to be extremely derivative.
Metalcore and deathcore (especially deathcore) are plagued with an image problem, but there are plenty of terrific bands if you know where to look. The more grindcore/mathcore end of the metalcore spectrum (bands like Converge) actually combine hardcore and metal instead of offering up breakdown-plagued abominations.
When it comes to good metalcore and deathcore, I suggest the following bands:
Converge (obviously), Botch, The Dillinger Escape Plan, The Contortionist, Shadow of the Colossus, Sikth, Lorelei and Ovid's Withering (if you can stomach an orchestral approach).