Learn Russian: На берегу реки появилась длинная велодорожка. - A long bike path appeared on the bank of the river.

In Russian, the most important new information usually goes at the end of the sentence. By putting велодорожка at the end, the sentence emphasizes that a new bike path appeared (translating naturally as 'A bike path' rather than 'The bike path'). Starting with Велодорожка появилась... would sound like we are discussing a specific path we already knew about.

Why is the prepositional form на берегу instead of the standard на береге?

Берег belongs to a special group of masculine nouns that take a stressed -у ending in the prepositional case when used with в or на to indicate physical location (this is often called the Locative case). Other common examples you have likely seen are в лесу (in the forest) and на полу (on the floor).

How is the word велодорожка formed?

It is a compound noun made by combining вело- (a common prefix for things related to bicycles, short for велосипед) and дорожка (path). You will see this same prefix in other everyday words like велогонка (bicycle race) or велоспорт (cycling).

Could I say велосипедная дорожка instead of велодорожка in this sentence?

Yes, велосипедная дорожка is perfectly correct grammatically and means the exact same thing. Велодорожка is just a shorter, more conversational way to express it, much like choosing 'bike path' over 'bicycle path' in English.