Here's some tips from my playthrough.

1. The vent system above your cell is everything. Spend as much time clearing the vent way as much as possible. It's the main route of accessing the right side of the prison and plans your escape way.
2. Be on your best behavior. NEVER attack guards or missed an appointment. This reduces the chance of guards checking your cell and putting down the curtains. This is obviously to hide the big open vent hole on your ceiling. You may attack them after you have a fake vent cover.
3. The only two items you really need from the interaction area are Magazine and Duct Tape (to make Poster). Don't waste too much time and money on it.
4. Find guard with red key and make copies of it. Ignore other keys (except Orange, but it's still optional) 'cause you won't ever need them.
5. The escape route is straight-forward. You may look at the whole map from the workshop to better understand the route.

That's all from me.

I escaped day 9 with no key required:

step 1: Get in vents using screwdriver.
Step 2 : never worry about open vent since you can put the vent back on (Otherwise they will go in and see it if you get cell searched anyway).
Step 3: climb through vent ladder and use forks to get to generater area.
step 4: go on roof and jump off using sheet rope (note that I wouldn't recomend doing this off the north side due to extra fences.
Step 5; Cut electric fence before generator turns back on

pro tip: the Cutlerary is your best friend and re-spawns with a fresh batch each day, make sure you are hoarding it in your room and vents:)

my tip is very situational as i got a stepladder from an inmate but if you manage to get one you can use the vent entrance in the small room besides your cell and make your entrance from solitary as the guards never walk into it but do place the wall back down when you break it, like i said, my tip is situational

and like the guy above me said, cutlerary is your best friend