This is a basic guide to that. It is not the same approach I use with in-person training and is intended instead to be a somewhat more conservative approach that is safer and easier to do without direct supervision.
When to progressWhen you can do more than the target number of reps for the last set on an exercise. Beating your target when you are fresh doesn't count. It's the last set that we are looking at. If the target was 10 or more reps then you need to do 2 more reps. If the target was between 5 and 8 reps then you need to do just one more rep. If it's under 5 reps then one more rep may be too big a jump - see below.
How much to progressFor major exercises, and in particular lower body exercises, then 10-20% is a reasonable increase. Major exercises are the compound movements involving multiple joints like squat and deadlift, and for some folks bench press.
For upper body exercises and ones that are single joint exercises, then 5-10% is reasonable. This includes single leg or single arm exercises, and accessory exercises.
Depending on equipment limitations then you may need to wait for more reps before going up in weight. If you are doing an overhead press with a 20 pound dumbbell for 10 reps, and your gym only has 25 as the next available weight, then 2 more reps probably isn't enough. You'll probably need 3-5 more reps before you can handle the increase. And you may need to drop down the number of reps the number of reps at first, maybe doing just 8 for a week or two at the new weight.
Slow is fineGoing up 5 pounds a week in your major lifts is just fine. If you actually kept that up you'd add 250 pounds in a year! Don't feel the need to rush. The strength will come if you put in the time.
For major lifts where you are in the 5RM or fewer reps range then just 2.5 or 5 pounds a week is just fine, even if you can't do one more rep yet. Maybe you hit that new weight and maybe you don't. If you don't then just drop back to the last weight. It's not worth stressing about - every week can't be your best week.
Progressing the Program
The above advice is geared towards progressing weight for an exercise while targeting the same number of reps. The program should also progress by increasing intensity, which means more weight at fewer reps. I discussed intensity levels, and progressing them in a previous post.
As always, feel free to reply with questions.