I have pain & swelling in tendons of my wrist and also my thumb caused by long sea kayaking trip 5 days ago. What to do to ensure it heals properly?

See below. Rest, ice (if there's swelling/pain) on & off throughout the day, elevation (for swelling), NSAID (advil/motrin/aleve/etc) for pain & inflammation (unless you're on blood thinners, have ulcers/bleeding problems). Gradually do range of motion exercises for wrist, fingers, thumb to keep them limber. If no swelling, can use moist heat instead of ice—whichever feels better & helps more.
Wrist pain. If in fact you have developed an acute tendonitis, the treatment(s) are aimed at reducing the inflammation/irritation. Above all else you need to rest the irritated body parts. Beyond that, if not contraindicated, you can use other anti-inflammatory measures including medication. Beware that if you use a splint/brace make sure it includes the thumb .