Blogs, review sites, last.fm and spotify (related artists feature is superb) and generally via word of mouth through the few people I can rely on to recommend me decent music. Spotify is so good, I even pay a fiver a month for the unlimited service - as much music as you want and no ads. Their catalogue is fairly extensive as well. I still try and get to as many gigs and buy as much new music as possible but with money being fairly tight at the moment it's more often a case of getting via other means... <_<
::Last.fm - Make an account, list all the music you listen to (or download the scrobbler which will track each song as you listen to it), and there's a section which'll recommend you more music. I've discovered loads of new artists through this, such as Emilie Autumn and Kat Deluna. I've also talked to loads of new people here, some who have recommended artists like The Jezabels to me.
::Youtube - Because I tend to listen to a lot of my music through Youtube, the recommendations bar on the side has given me loads of new artists to listen to. One of the most significant is Kerli, who's an artist I constantly listen to. Her song 'Walking On Air' appeared on the sidebar, and I fell in love with her voice, musical style and visual style, leading me to import her album. It's also led me to listen to other similar artists like Iiris.
::iTunes - If I'm looking at an album on iTunes, the "You May Also Like" bar gives some very good albums to listen to. Through this, I've discovered a variety of artists who I've listened to and ended up loving. You can also look at the Singles/Albums charts for other good ideas (it gives a decent mix of mainstream and more obscure choices, but more of the former). The free downloads are also pretty good. For instance, one free download was 'Home' by Gabrielle Aplin, which is a beautiful song.
::TV - Stuff like Jools Holland is excellent for showcasing brilliant music, I've found. You can also find out artists through other shows like Never Mind The Buzzcocks (which is where I discovered Paloma Faith years ago, and Do You Want The Truth Or Something Beautiful? became one of my favourite albums). Also, many adverts will showcase songs which end up becoming more popular, as recently seen with Too Close by Alex Clare. You could download Soundhound or a similar app for your phone, which'll listen to a song that's playing and tell you what song it is.
::Music Blogs - Often, these sites offer newer/more indie choices, but many of the artists they'll talk about are usually excellent and often do stuff such as free song downloads and mixtapes. For instance, one blog talked in detail about an artist called Queen Of Hearts, who gives off a Goldfrapp/Rachel Stevens vibe, which works well.
::Gigs - There's been times where I've been to a concert and I've not been a fan of the support acts (e.g. wasn't a big Devlin fan when I saw him support Jessie J), but often you'll find a decent artist. For instance, I went to see Alexis Jordan and Loick Essien was supporting. I thought he was excellent, because he's got a brilliant live voice, and he really engages with the audience.
::Word Of Mouth - Ask your friends for new music! I asked one friend if he knew any good jazz artists, and he recommended both Diana Krall and Melody Gardot to me, both who are fantastic! Another friend suggested The Band Perry to me, and I really got into their music. I'm even seeing them live soon!
::Shopping - If you walk past a nice music shop, feel free to pop in and have a look at the CD's. You never know which albums will catch your eye! You might even discover an artist you've not heard in years and buy their album to relisten to the tracks or to discover new tracks. Working in a charity shop (specifically with music and DVD's), I'll come across a lot of new interesting artists, which I'll end up going home to listen to in more detail. I might also find an artist who I know but who I've not listened to in much detail, and ended up putting the CD on and loving their music (Ella Fitzgerald comes to mind).