On clear sunny days, cottontails can usually be found basking or grazing on the south side of hills or riversides.
I've had the best luck in mixed forest. Their first instinct is to stop moving so that they are not spotted. Failing that, they will bolt, sometimes only after nearly getting stepped on.
Kicking up brushpiles & whatnot is usually productive. Usually best with a partner to take the shot while you're flushing the critter out.
Briars & thickets are good too. It's nearly impossible to be the flusher & the shooter at the same time, so a partner is best. Place your shooter on the opposite side of the thicket whilst you plow in, call any flushes, and wait for the report.
Obviously when shooting with a partner, brief on fields of fire, and who's shooting & who's flushing. Best to walk line abreast in more open forest. Thicker forest may require the flusher/shooter method.
Since I was 16 I've been blasting bunnies at my friends farm. If I put a notch in my mossberg for every bunny slain, I probably wouldn't have a stock left
My preference is #5 or #6 shot with an improved or modified choke (depending on vegetation density). I've tried #7 1/2, but most pellets fail to penetrate adequately. Moving up to #4 or #2 shot is effective, to be sure, but it also bruises a lot of the meat.
For years I hounded the old man to come out with us, and he always had excuses. Finally, one day was perfect, fresh snowfall, clear skies, -10C. I told him no excuses, you're coming with us. Wouldn't you know it, the old bugger outshot the two of us.
Happy hunting!