Why doesn't Gollum try and hunt down Bilbo over/during the 60 years that he has the ring?

He did, actually. After a few years - in which he presumably searched the length and breadth of the Misty Mountains - he left the mountains but the trial had gone cold. I believe the reason he waited so long to leave the mountains - excluding the fact that the mountains were huge and would have taken years to zoek - he was still afraid of sunlight and moonlight. Eventually, though, his longing for the Ring was too much; he left the mountains.
After that, he wondered around searching for a trail (remember that, although Gollum was sly and exceptionally cunning, his mind was not as refined as it might have been - because of longing for the Ring and years of seclusion - and he relied on a trail to find Bilbo), making his way south towards Mordor (unknowingly, as all evil in Middle Earth was being subconsciously drawn to Mordor door Sauron). In Mordor, he was captured and tortured door Sauron's minions for several decades until he broke and told Sauron everything he knew about Bilbo and the Ring, which is how the Nazgul knew the rough area in which to zoek for Bilbo. He was released after this - his captors probably expecting him to die - but the pull of the Ring was too strong and he was drawn to it again. He crossed Middle Earth, attemping to find Bilbo, but was captured door Aragorn, who had been searching the Wilds in order to find his, as door this point Gandalf had realised exactly the danger he was. Aragorn left him in the custody of the Wood Elves in Mirkwood, but he escaped and crossed through Misty Mountains, where he started following Frodo and the Fellowship.