I understand that in order to get results out of a workout one must invoke progress overload into the workout. One way of doing this is to have less rest time, and another way is to increase the amount of weight being lifted. My question is though, if say i do 2 sets of bicep curls with 25 lbs dumbbells for seven reps is there any benefit of doing another set or is it simply better to go up in weight or cut my rest time? Some things i feel i should point out to help you answer. Firstly my barometer to know when its time to go up in weight is when i can do 7 reps on my last set (i know some of you may do more reps per set before increasing weight, but i am currently trying to get stronger hence the low rep range). Another thing i should point out is that i do not only do 2 sets of bicep curls i also do 2 sets of hammer curls for a total of 4 sets. With that in mind is there any benefit to adding another set?
Since your goal is strength, then definitely up the weight as frequently as possible, keeping the reps in the 3-5 rep range (7 is a little high). Rest time should be whatever is needed in order to do the next set properly. If you cut rest time, you're training muscular endurance rather than strength.
Adding a set will provide extra stress, potentially triggering more adaptation of the muscle, but would also require more recovery time. When you find that 4 sets of 7 reps is insufficient to stimulate the strength gains you want, then you can consider adding additional sets, but add 2, rather than 1, so it's actually a significant difference, keep them in the low rep range, and rest sufficiently between sets so that you're actually able to do them. However, if you add extra sets, you may not be able to lift as frequently as you are right now, or the workout following your extra-set day should be a lighter day.
This guidance is based on material from Mark Rippetoe's Practical Programming for Strength Training, specifically, Chapter 4 (The Physiology of Adaptation), and Chapter 7 (The Intermediate), where he talks about sets, reps, and the Texas Method.
It may be easier for you to make smaller, more easily predicted increases in weight by switching to barbell curls as your main bicep lift. That will allow weight increases as small as 5lbs (or even 2.5lbs if you buy some 1.25lb microplates). Distributing that weight increase across two arms is much more doable than constraining yourself to the increments available in the dumbbell rack, especially since those are single-arm exercises. One arm getting stronger by 5lbs takes much longer, and causes a much greater change in number of reps when you make the jump than your two arms getting stronger together by 2.5lbs. See this article for more motivation behind maintaining incremental weight increases.
As an example, pretend your arms are individually strong enough to bicep curl 20lbs.
If you're doing dumbbell curls, and wait until you can do 7 reps with 20lbs, when you jump to 25lbs dumbbells (a 25% increase in weight), you may have to drop down to doing 3 reps, because that's a huge jump for such a targeted exercise.
However, if you're doing barbell curls, and you are doing 5 reps with 40lbs, jumping up to 45lbs (a 12.5% increase in weight) is half as difficult a jump that would have happened with the dumbbells, and your reps might drop to 4 or 3, but you'd be back to up 5 reps pretty quickly. If you find microplates, you can increase the loading even more gradually.