I would say 8-12 reps. Less than 8 reps would be better for strength gains, but I think 8-12 is better for definition. It depends on what you would like most You should still get strength on both though and definition (provided your body fat is low enough)

Start on the weight where you can perform the exercise with correct form for around those number of reps. The last 2-3 reps should be a struggle but will not losing form.

Push ups can be worked on by doing "girl" push ups (basically normal push ups but on your knees instead). If you do other chest/shoulder/triceps exercises it should improve strength there too.

Definition in your arms can be created first by a low enough body fat. It doesn't matter how much you train there, if you still have excess fat, your hard work will barely show. To do this you need a calorie deficit to reduce overall body fat. Assuming you are already OK in this area. Bicep/Tricep and shoulder exercises will help give you nice defined arms

Chest: Push ups / dumbell bench press / dumbell flyes
Biceps: Bicep curls / chin ups (palm inwards) - but you would need a pull up bar for that.
Triceps: Skull crushers (youtube should have demonstrations) and another where you lean forward and push the dumbell back underneath you - but I can't remember the name

first how is everyone this morning?
so i have 10 and 5 lb. free weights here at home. what do u recommend the reps be to see some results. im not really new at it but i never have actually stuck with it long enough to get good results. two things i would love to be able to do: one being push ups. can't seem to do them to save my life. and the other is i would like to see some definition in my arms. those are my goals.

smb0701: You should consider joining the 'Hundred Pushups Challenge' group here on MFP. Any pushup variation is acceptable (on the knees, off a bench/sofa, etc.), and even though 3 pushup workouts a week are listed, doing 2 of them per week would be a great start (or doing all 3 workouts, but less sets per workout).

It's just about doing something towards your goal and having group to support and motivate you when you need it.

Good luck and hope to hear about your pushup progess!