This heavy version of the Tears for Fears classic is very good, and a solid exercise for developing your chugging hand. Use Drop D tuning - i.e. tune the Low E string down to match the open middle D string. You only need detune that Low E string.
This song is not so much about speed technique as it is about tight control and consistency. The riffs are not difficult at all - especially at slower speeds. This means you should be able to learn all of them and switch between them quite quickly. Once you've mastered each song part slowed down, the rest is just speeding up and keeping tight with the drums.
Disturbed use a lot of tight rhythmic riffs, this means great practice for both hands, particularly your strumming hand. You will need to alternate between Open chord stabs and palm muting. Don't forget to mute with your fretboard hands between chord strikes too. This keeps the guitar punchy and generates a stronger, tighter feel to the rhythm.
Verses - Start each bar with an open strike on the powerchord, followed by palm muted down / up strokes:
Bb5 (stab) - d u d u d - d u - d u - d u (repeat)
The dashes are brief gaps in the rhythm, equivalent to 1 pick stroke.
Listen to the rhythm and watch the pick strokes. There are gaps in the rhythm, so it's important to listen first. Get that rhythmic pattern ingrained in your head, then start playing it. The more familiar you are with the picking rhythm, the easier it will come when you start to practice it because you'll know how it sounds and when you're doing it right.
The picking pattern is short and repeats for every chord the same way during the verses. Then punch out the Eb5 chord using palm muting and down / up strokes with slight gaps between them again. If you can play the first part already then this will be easy to pick up.
Choruses - Just nice and simple chords, even if there is a little stretching involved. End with more chugging on the Eb5 chord (this time use open strikes not palm muted, and don't forget the cool slides!)
Bridge - Easy 6 note loop, repeating after the slight bend. End on the Eb5 and Bb5 then hit the solo.
Solo - This is really just another repeating lick, and not too difficult. Just a little higher sounding than the previous song parts.
Outro - This is a variation on some of the earlier verse riffs and should be straightforward enough to get. However, at the end is another riff change which involves a quick harmonic. Skip that if it is too difficult. Disturbed love throwing in harmonics at awkward points in their rhythms, and just couldn't resist here. It sounds nice, but you don't need to worry about it if you can't do them yet - it's quite an advanced technique to quickly throw them in amongst riffs!