There are some physical signs of illness you could start to look for if you are worried that your betta is sick:
Unusual red splotches (bacterial infection)
White and fuzzy growths (fungal infection)
Fins curling (infection/fin rot)
Fins darkening (fin rot)
Fins receding (fin rot)
Unfortunately these have all been learned through experience (purchased sick). I would be happy to share how I treated these if you see any of these signs.
There are other signs of illness as well that I am less familiar with. Hopefully someone else can help you with those.
If you don't see anything physically wrong and you've just added your betta to your tank, he/she may just be stressed because it's in a new environment or the filter may be too strong. If your don't see your betta exploring after a week, start looking into ways to lessen the strength of your filter.