From the front, it's about half a mile ... out of the main exit from the station, straight down Queen's Road (stopping in at the Seagulls Supporters Shop, half way down on your left). Keep going down West Street ...
... avoid turning right at all costs ... you will end up in Hove !!! ...
Easy.
But you'd probably want to end up a bit more centrally. Going down Queen's Road, turn left into North Road, all the way to the bottom, and then right, taking you past the Pavilion. That way, it's nearer a mile, maybe a bit less.
But for shops, work your way down North Road, and take any of the small roads off to your right. That will take you into The Lanes. Keep on going, and you end up at the seafront again.
The thing to bear in mind about Brighton Station (which you won't see if you just look at a map) ... is that going back to the station from the shops or seafront ... it's ALL uphill ... and quite steeply uphill !!