The Lord of the Rings leads us to believe that he hadn't travelled further afield than the Shire.
However if his friends are to believed, he had been to see the Elves:
All quotes from: The Fellowship of the Ring - Book 1, Chapter 2: The Shadow of the Past
Frodo went tramping over the Shire with them; but more often he wandered by himself, and to the amazement of sensible folk he was sometimes seen far from home walking in the hills and woods under the starlight. Merry and Pippin suspected that he visited the Elves at times, as Bilbo had done.
However, further on into the same chapter, we're led to believe that he'd never left the Shire:
He found himself wondering at times, especially in the autumn, about the wild lands, and strange visions of mountains that he had never seen came into his dreams. He began to say to himself: ‘Perhaps I shall cross the River myself one day.’ To which the other half of his mind always replied: ‘Not yet.’
While it is unclear whether he left the Shire or spoke to Elves, he certainly got close to both, by "walking and talking with the strange wayfarers" who were abound in the Shire at the time.
Frodo began to feel restless, and the old paths seemed too well-trodden. He looked at maps, and wondered what lay beyond their edges: maps made in the Shire showed mostly white spaces beyond its borders. He took to wandering further afield and more often by himself; and Merry and his other friends watched him anxiously. Often he was seen walking and talking with the strange wayfarers that began at this time to appear in the Shire.
The strange wayfarers may simply have been the Dwarves passing through the Blue Mountains:
There were, however, dwarves on the road in unusual numbers. The ancient East–West Road ran through the Shire to its end at the Grey Havens, and dwarves had always used it on their way to their mines in the Blue Mountains. They were the hobbits’ chief source of news from distant parts – if they wanted any: as a rule dwarves said little and hobbits asked no more. But now Frodo often met strange dwarves of far countries, seeking refuge in the West.
However, in those times the Elves were also abound, walking the old East-West Road to the Grey Havens.
Elves, who seldom walked in the Shire, could now be seen passing westward through the woods in the evening, passing and not returning; but they were leaving Middle-earth and were no longer concerned with its troubles.
The lack of concern of the Elves with the troubles of Middle-earth may suggest that they were unwilling to speak to strangers, furthermore, Frodo's first meeting with Gildor and his company he seems surprised that they were aware of his name. While Frodo may have been aware of the departing Elves, and therefore any Elves he'd met wouldn't have gone back, he may still have expected news of the nephew of Bilbo's wanderings to have made it back to other ears. Gildor later states they new of Frodo having seen him and Bilbo wandering, although it is likely he hadn't seen them.
And how do you know my name?’ asked Frodo in return.
We know many things,’ they said. ‘We have seen you often before with Bilbo, though you may not have seen us.
The Fellowship of the Ring - Book 1, Chapter 3: Three is company
He did, however, know where one could find Elves, suggesting he may have met with them, and given he had knowledge of some words spoken in the tongue of the High Elves, they may have taught him something.
One can meet them sometimes in the Woody End. They don’t live in the Shire, but they wander into it in spring and autumn, out of their own lands away beyond the Tower Hills...
It was singing in the fair elven-tongue, of which Frodo knew only a little, and the others knew nothing.
ibid.
