Alternate picking is hard. Once you get the hang of it, a lot of things become much easier to play.
How do you get the hang of it? Start with turning on a drum machine or metronome, play a basic pattern slowly and accent each UP stroke as you play. Really make the UP pick LOUD and hard. Speed up gradually until you start to ‘feel’ the up stroke just as strongly as the down stroke.
An important tidbit is to keep your right hand/wrist very loose and make sure you keep the same ‘posture’ in your hand with down and up strokes. Economy of motion becomes more and more important as you speed up.