For soloing, you need to practice
alternate picking and scales a lot (woodshedding). It is a lot different than practicing the "song" which I assume by that you mean the
chord progressions.
It can take years to get good at alternate picking depending on your diligence and other factors. Practice/learn the pentatonic scale(s) and the blues scale but Mr. S hit it on the head, the modes are the holy grail as far as I'm concerned for soloing.
I've been playing 25 years and I took lessons only for 5 months when I started. I learned the basic blues scale but practiced alternate picking until I got good at it. Then I learned some of the pentatonic stuff. I could play (ape) most anything I wanted rhythm and solo. But I was never good at jamming/soloing because I didn't "get it". I could burn in one position blues or pentatonic, but how to move around in key - I was lost.
Around January of 2005, I bought a book/DVD called "Shred is not Dead" by Terry Syrek and first learned about the modes and I could finally understand what I had been missing all along. Another excellent book is "The Exercise Book" by Adam Kadmon which has all the modes in it with tons of exercises involving them. Like Pawel said, lessons from a good teacher might be a much better way to go for you. A good teacher can help you to not pick up any bad habits on the way and help speed your learning if that's what you want.
The modes are sometimes referred to as 3 note per string scales and there are 7 of them - Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, and Locrian. If you practice the heck out of them it will not only help your alternate picking but help you learn theory and how to move from one mode to the other up and down the fretboard.
Another thing to consider is buying some kind of jam tracks to practice your soloing in different keys. Another book "Shred Guitar" by Paul Hanson has not only lots of good licks/sequences to practice but also has a CD with lots of jam tracks on it. The Fender G-DEC amp is also great for jamming practice but it will set you back about $269. But you can practice in many different styles with it and change each one to any key you want. And it's not a bad little amp (but definitely not great) to boot.