Advice: Learn/google proper "economic picking". Ideally you'll want to pick as many notes as possible and learning these techniques will help you strike your starter notes when you do actually want to "slur" more effectively so the hammer on/pull offs ring out properly.
When slurring, and in general, you need to obey proper fretting technique with your left hand. Keep the tip segment of your finger at as much of a downward angle toward the fretboard as possible. This means the last nubbin' of your finger should be as close to a right angle in relation to the fretboard as possible and make sure the TIP of your finger is what is being pushed into the fretboard. Not the pad. I see this very often where someone is not using the tip and it leads to sloppy fretting, poor/uncontrolled bends/vibratos and it generally slows the player down. It also mutes the other strings unnecessarily so they aren't as accessible when they need to be. This can be difficult (and sometimes painful) at first. The natural instinct is to use the heel of the thumb as the counterpressure on the back of the neck to successfully fret notes but this restricts the fingers from moving freely and will make people straighten their left hand fingers out which means they aren't fretting with their fingertips... they fret with the pads. The only time you should be using the pads of your fingers is when fretting more than two strings at a time (like a barre). For proper counterpressure from the back of the neck for fretting single notes/doing leads do NOT use the heel of your thumb. Use the pad of the thumb above the second knuckle (the last segment of the thumb where the nail lives). This will force the palm of the hand off the back of the neck of the guitar and allow the fingers to curl toward the fretboard so you can use your fingertips. If this is not how you already play it will be foreign, weird and possibly hurt but the muscles will develop if you stick to it and practice some scales this way. As you move between the top to bottom strings allow your thumb to change position so you are getting the best counterpressure. It is difficult to describe in words but it should become apparent as you play scales or arpeggios across the strings.
Also keep your fingertips as close to the fret as possible (without actually being on top of the fret). The closer you are to the fret the clearer the note will be. This is particularly important when "slurring" (hammer ons/pull offs). If you "hammer on" too far away from the fret it may not cause the string to vibrate as aggressively as it could making it sound dull. If you intend to "pull off" from that poorly vibrating string the second note may not sound good either because the string is already going lifeless (unless you aggressively do your "pull off" but then that sounds unnatural, takes extra strength which might slow you down for the next note in the passage and is easily avoided by simply not allowing the bad fretting to occur in the first place). In the case of pull offs, if the second note isn't fretted properly then again you will end up with a dead sounding note... in fact likely more so than the hammer on note.
The point? Developing your technique so that you are a) using your fingertips and b) keeping those fingertips as close to the fret as you can helps make your playing more defined (which IMO opinions is far more important than playing fast) and helps with speed because of the natural design of the human hand (ie: it moves faster that way and forces the palm of the hand off the back of the fretboard which is very limiting), it is more ergonomic, it allows hammer ons/pull offs two be properly and efficiently executed and it frees up the mind from the mechanics of playing to focus on your next move (instead of forcing your hands to muddle along and getting frustrated at the tone being produced).
Both of those also help immensely with sweet sounding bends and vibratos (however with vibratos you get a little more leverage if you move the string SLIGHTLY more toward the pad of the fingertip... that technique can be viewed by watching BB or Albert King videos).
One final note... practice scales. Get a good scale book and practice the ever bleeding poop out of them with and/or without a metronome. It really is the best way to develop speed and agility.
Cheers.