Power metal is a faster and sometimes more aggressive offshoot of traditional heavy metal that started in the mid 80s. It's guitar-driven, more melodic and less violent than thrash, although there are plenty of bands that have elements of both. Not my thing as a general rule, but I enjoy Demons And Wizards and the first Sanctuary album, and I guess early Queensryche and Metal Church could qualify. I liked Manowar when I was 12 or 13, but don't tell anyone... I don't get into symphonic PM at all, but I'm sure someone here could give you good recommendations.
"Symphonic" isn't a genre, it's just another adjective to help narrow things down. In practice it usually indicates a lot of sampled strings and piano sounds, but sometimes the bands use real classical instruments. So you can have symphonic power metal, symphonic black or death metal, etc etc. I've been listening to a lot of Septic Flesh, Diabolical Masquerade, and early Cradle Of Filth lately, check them out if you're interested in heavier music with symphonic elements and harsh vocals.