和 = and (derived from harmony, the way people should be with each other)
跟 has the implication "to follow"
与 is the literary form. However, there are four other meanings (used mainly in literary Chinese)...
1. "to give" 与人方便, be considerate of others (literally to give convenenice to others)
2. "friendly relationships" , "good friends" 相与, 与国 (friendly countries), e.g. 今时朝鮮与美国非与国也 N. Korea and the US are not friendly countries [to each other]
3. "to praise", "to laud", 与人为善
4. As a conjunction 与其used with “不如”、“宁可". Sometimes it can be translated as "it is better than". "与其殺是僮，孰若賣之" It is better to sell the slave boy than to kill him.