Come up and browse the web with expert (7)
I believe the answer is:
surface
'come up' is the definition.
(I know that come up can be written as surface)
'browse the web with expert' is the wordplay.
'browse the web' becomes 'surf' (I've seen this in another clue).
'with' is a charade indicator (letters next to each other).
'expert' becomes 'ace' (ace can informally mean very good).
'surf'+'ace'='SURFACE'
'and' acts as a link.
(Other definitions for surface that I've seen before include "Become apparent" , "In water, come to the top" , "Top part; outward appearance" , "Outside part" , "emerge?" .)