It's difficult to make a single recommendation. For a start, it is going to depend significantly on your current level of fitness. And as you get stronger over time, the weight you should be using should go up.
Also, it is going to vary significantly depending on what exercises and muscles you use. For example, for dumbbell lateral raises (small muscles holding the weights a long way from the body) you will likely manage only a few pounds, while for something like shoulder presses (larger muscles closer to your center of gravity) you should be able to manage something much heavier.
The real object with strength training is not a specific weight, but to ensure you are genuinely challenging your muscles. This is generally taken to mean a weight heavy enough to fatigue your muscles in 12 reps or less. If you can do more than 12 reps, it is time to move up to a heavier weight.
As your strength will increase significantly if you strength train regularly, you may find an adjustable set of dumbbells better value than a full set of fixed dumbbells.
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