With the dumbbells you've already got, you're halfway there, honestly. Bodyweight alone lets you do tons of stuff: push-ups, pull-ups if you have a bar, squats, lunges. The thing is, it's not really about what equipment you use - it's about stressing the muscle with a load that forces you to work hard. If the dumbbells you have are heavy enough to make the last reps of a set difficult, that's perfectly fine. Ideally you'd alternate compound exercises (ones that work multiple muscles) with isolation work when you can: like dumbbells for shoulders and arms, push-ups for chest and triceps, bodyweight squats or dumbbell squats for legs.
For frequency, three or four days a week is a solid compromise, especially starting out. You don't need to kill yourself every day - actually, if you hit the same muscle group too often without proper recovery, you won't grow well and you risk injury. You could do one day for shoulders and arms, one for chest and back, one for legs, then repeat. Between sessions for the same muscle group, wait at least a couple days. The thing most people skip is eating enough and sleeping well - without that, your muscles won't grow even if your training is perfect.