The first two Gatherer achievements will come very quickly through standard play. However, hooking 5000 items takes quite a long time. Just remember that picking up items from collection nets and mining items underwater both do not count towards this set of achievements. Barrels also only count as one item hooked, rather than however many items the barrel may contain.
Plastic hooks are recommended while grinding towards Expert Gatherer as it will save you a lot of resources.
These achievements do track over multiple worlds.