What would cause a chain to jump gears on take off and how can it be fixed?

Normally this would indicate a tension or location issue.
What type of bicycle is this ?

Or it could be a dérailleur alignment issue. What dérailleur do you have?

Deraillieur or the chain is stretched and worn... or both.

The first thing to check (and most likely reasons) is a dérailleur adjustment. Especially if you have indexed shifters (that click to change gears). On the dérailleur are tensions srews to adjust the tension of the gear cable. You can play with that adjustment in tiny increments until the shift works smoothly.