For most people chest and arms are exercised on the same day, often dividing days between upper body and lower body. If all I had were dumbells, I'd focus on dumbell deadlifts, one arm standing overhead press, and arm curls.
For a while I did exactly that, using Pavel's book "Power to the People! : Russian Strength Training Secrets for Every American" as guidance. As for how many reps/sets to do, that depends on whether you're trying to build strength and/or size.
Example: For strength stick with a couple sets of 4-8 reps, never lifting to failure, for both do several sets of 4-8 reps, or for normal bodybuilding with its emphasis on size, do perhaps 3-5 sets of 8-12 reps, more like you seem to be doing (there's a bit more to it than this).
Good luck, and another excellent, more mainstream weightlifting book is former Mr. Universe Bill Pearl's "Getting Stronger". It has a huge selection of exercises and general advice for beginners and experts. Now in his 70s, he's been a vegetarian for decades and still lifts weights quite a bit.
As for deciding if you ever need to add a day between workout sessions, if you find yourself hitting a plateau and not getting any stronger/bigger, consider adding a day of rest/aerobic exercise. Lifting weights every other day is usually plenty.