i'm new to electric...

Any kind of solo, it sounds like i'm pulling off, cause the string rings while i go onto a different string.. for example..

after i pick that 4th fret, 5th string... it rings while i go to the 3-6.. because i guess i'm pulling off too fast? but how can you play fast without pulling off?

should i be trying to mute the string (with my right palm) after i hit it?

one more thing.
when i'm strumming chords.. should the other strings be ringing too?
i strum a D.. just the bottom 3 strings.. and then lift those 3 fingers off, as if to stop the sound, but the other 3 strings still ring so loudly.. that well, its annoying.
same thing if i do a power chord on the 5th, 4th, and 3rd strings... the 6th string is still ringing loudly.. even though i didnt touch it... quite annoying

You're not going to be able to do this stuff perfectly straight from the off, you're going to have to work at it for a while.
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should i be trying to mute the string (with my right palm) after i hit it?

or the finger on your left hand, but that's harder to do on fast stuff
Rockin' shit
what song is that from?
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what song is that from?

i'd do anything - simple plan
As for your chord question. No. Unless they are notes within the chord. And the D you need the open D string ringing plus the bottom 3, unless you doing an inversion in which D wouldn't be the lowest note.
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for your solo question, just slow every thing down...just because you can play the notes fast in the correct order, doesnt mean you should....to start off with. Play it really slow until you have everything absolutely nailed how you want it to sound. Gradually increase speed until its perfect, and there you go. It will sound lame at first but helps alot.
you dont even have to lift your fingers much to play this lick.
For example, 4 with your middle then 3 with your index and 6 with your pinky, then lift just lightly enough to stop the noise and move up 2 frets
Stop whining and learn your theory!

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I know this song, and gradually I played it fast, this is one of the easier ones, mind you, I play rock songs on Acoustic, too cheap to get an Electric.

Practice like the others said, it takes time, I did it slowly, and you should be able to figure out the finger placements, that suits it.