I love the blues because there's so many ways to approach improvising over it. I'll offer up two of my favorite approaches with this caveat: these scales, modes, and chords feel a lot like rules and when it comes to music (and life actually) my mantra is this:
- Follow the rules
- Break the rules
- Make the rules
So in other words, learn to follow these guidelines, then try throwing in some different notes here and there, and even switching between the different frameworks, and eventually (down the road) throw it all out the window and make your own rules.
I'm also assuming the chords are A7, D7, E7. For minor blues where the chords are Am, Dm, E7 (or Em) I would have a totally different answer.
Approach #1: blues scale
This one's simple: play the A blues scale over all three chords. You probably know this already but in case not the blues scale in A is:
A C D D# E G
So it's just a minor pentatonic with the D# passing note thrown in. Using just this scale you can get so much mileage it's not even funny. After all, B.B. King, Albert Collins, Albert King, &c &c played almost nothing but those 6 notes their entire career.
Of course you don't just play those notes up and down in sequence over and over. That would sound awful. To avoid that repetition learn to rest on the right notes for the right chords.
On A7, rest on A, C, and E.
On D7, rest on A, C, and D.
On E7, rest on D, D#, E, and G.
And since this is the blues you'll want to be bending notes liberally. The most common bends to use are (on the top 3 strings):
D to E
D to D#
G to A
If you want to spice it up a bit, try throwing in F# over the D7 (F# is the 3rd of D) and B over the E7 (B is the 5th of E).
Approach #2: jazz blues feel
Here you play the mixolydian scales (or modes to be accurate) over the A and D chords and play the blues scale over the E chord.
So over A you play:
A B C# D E F# G
Over D you play:
A B C D E F# G (note the only difference is C vs C# here)
And over E7 you play:
A C D D# E G
I'm not even getting into rhythm because that depends so much on the song, suffice to say rhythm is crucial. With bad or boring rhythm, great notes sound awful. With amazing rhythm, almost everything sounds great no matter the notes. For now just think about mixing things up. If you're in 4/4, try grouping your 8th notes into groups of 3, 3, and 2 (adds up to 8 for a full measure) or 3, 3, 3, 3, and 4 (adds up to 16 for two full measures).
Hope that makes sense and is helpful. Good luck!