Rules-as-written, you retain your existing languages and can understand the languages of your new form, for the duration of Wild Shape.
Rules-as-written, you assume the statistics of your new form, which includes languages. (A DM may find this illogical and is free to rule otherwise.) You retain your class, race and other features, which includes languages.
Gaining the languages of your new form
As per the rules on Wild Shape:
Your game statistics are replaced by the statistics of the beast, but you retain your alignment, personality, and Intelligence, Wisdom, and Charisma scores.
As per this question, a creature's "statistics" include the Language section of their stat block, as defined by Monster Manual p.6-11.
Retaining existing languages
You also retain your existing languages:
You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so.
All races in the Player's Handbook gain certain langauges, including Common, as a feature of their race. The source of these languages is explicitly stated to be one's race, (PHB p.17, under Languages):
By virtue of your race, your character can speak, read, and write certain languages.
For example, under Dwarf Traits, p. 20:
Languages. You can speak, read and write Common and Dwarvish.
The dwarf's ability to speak both Common and Dwarvish are explicitly stated to be features from their race, and would therefore be retained in Wild Shape.
The same specifically occurs for languages gained by class (e.g. a druid learning Druidic, a cleric of the Knowledge domain, a ranger's favored enemy). "Other source" is not strictly defined, but I would not find it unreasonable for this to include languages gained through background or training.
Jeremy Crawford also confirmed that a druid in Wild Shape retains the languages they know, and can speak them if their new form is able to speak.