How do people travel? When are they allowed to travel?The Giver
Besides the people who were appointed to have contact with the outside world, sometimes the schoolchildren from the Community are permitted to visit surrounding villages, but these occasions are rare. Jonas' sister Lily, for example, mentions having had a dispute with one of the children from another school, apparently part of a "field trip" to the Community. That evening she discusses the confrontation during the family's 'sharing feelings' time and ends up feeling a bit ashamed for her inability to put herself in the place of a child unfamiliar with the customs and ways of the Community. This is incidentally the one and only circumstance showing the link between experience and empathy, so important in the second half of the story.
As to transportation, bicycles are indeed main way of getting around, and it is the best means for Jonas to covertly escape with Gabriel during the annual Ceremony of Twelve. However, at the end of the story (when in the snow the bicycle is no longer useful) he finds a red sled, which he mounts with Gabe and then descends a hill to voices below. Jonas has already "experienced" this scene in one of his earlier dreams, which at this point the reader realizes was a projection of a future event rather than a memory of the past.
In 'Gathering Blue' (not exactly a sequel), the red sled is kept by Jonas' new community as a symbol of their leader's providential arrival and the hope his coming to their village has instilled.
In the book "The Giver" the only modes of transportation that I have read about has been by bicycle. The other only mention I could find was that the hills were all made level so that transportation would be easier. The book talks about the community establishing climate control to make growing for easier. (Section discussing this issue can be found on pg.85)
Reading further in the book Jonas makes his escape with Gabriel. He uses a bike for transportation.( Section discussing this issue can be found on pg. 163) There is also a mention of planes flying over head, but there is no evidence to support them doing any more than surveillance.