Translation:The groups travel to both America and France.
'begge' indicates two ''related'' entities - so 'both (specific, pre-known) two', rather than 'both this, AND this!'. If you're asking a friend with divorced parents what they're doing on Christmas: 'Skal du besøge begge dine forældre?' ('Are you visiting both your parents?'). However, if you've just described your holiday plans, someone might have to ask: 'Skal du både til Rom og Berlin?!'('Are you going to both Rome and Berlin?!')
If you're from Europe, the term "America" is mostly used as a shorthand form for the United States.
Mostly, yes. Us Europeans rarely make a difference there and we usually call everything collectively "Amerika". If you want to travel to both North and South America, you just say so: "til både Nord- og Sydamerika"