One of the biggest mistakes a newbie would make is that they try to rush through their progress, which usually leads to poor progress and in lots of cases injury. Don't rush, you're not just building up on strength and muscles... you also have to build up on your joints, tendon and ligaments. These last three take time to condition, so don't rush to injure yourself.

For pushups, i guess it's fine to go to failure once in a while but like i said don't got to failure at the expense of form. I would also say do it if you're going for a PR but if it's just for normal workouts it's not necessary.

As for Pull ups try not to go to failure if you're new to it. Pull ups are probably one of the hardest things to progress up as you're pulling up on your full body weight unlike a push up. Building up a pull up requires endurance too not just strength especially the forearms (which a lot of people neglect). It would be helpful if you gave more details on what you're working on.

I am able to do strict pull up of 10 reps for 2 sets with long rest in between (I do other workouts) and 5 reps for subsequent 2 sets. That is on good days. Less on bad days.

Another thing I notice is that if I were to do pull up every 48 hours, my body can still feel the fatigue. I can perform better with a longer rest interval but not doing enough can slow down my progress.

If you want to do supersets then do it right. 1 set pull ups, then 1 set whatever other workout you are doing (i'm assuming push ups), then back to pull ups again,.. etc. Don't leave too much time rest time in between them, take a few secs to catch your breath then move on to the next one.

Of course i don't recommend newbies doing supersets though, it'd be much better to focus one thing at a time. Build up on that endurance of yours... at the end of the workout, do some hangs, do some grip work... Sure you're tired from your sets but grip work is essential! Don't forget them!

Lastly if your body if tired, just rest. You don't have to be so rigid with your workouts, be flexible. If you must rest to get better returns then rest cause fatigue builds up eventually.

Will post progress in another month.