A good turkey choke will help your patterns 10-fold. There are a number out there to choose from and opinions on them will vary greatly. I've found that a good quality, extended choke tube in a .665 constriction works well with an 870 3" shotty. I don't think any state mandates you shoot steel only. If you decide to shoot lead, I'd try the #5 pellet size. The size of a #5 lead is ideal for turkey, ethically, out to about 40 yards.

Speaking of 40 and since you asked... that would be my max suggested shot on a turkey. Pattern at 40 and dial in your chosen shell / choke combination there. As far as desirable pellet count? Every one in the payload impacting the turkeys head and neck. Of course that's not going to happen, so I'd want an evenly distributed pattern that has * at least * 6 pellets in the turkeys vitals at 40 yards. That was the NWTF standard, but with todays Hevi-shot loads and high quality chokes, that amount of pellets is pretty much considered poor.

If I can't get 15+ pellets in the brain and spine at 40 paces, (not just the outline of the turkey head / neck), then I move on and test another load. You'll do well to read as much as you can here and sort out opinions. Watch for those who give reports on their patterning results. I don't want to confuse, but keep in mind no two guns, even of the same manufacturer, are going to pattern the same.

If it were me, having owned a 870 myslef that was a 3" gun, I'd buy an extended choke tube rated to handle Hevi-shot in a .665 constriction. I'd pattern it with #5 shot first, if lead was my shell of choice. Try some from different companies, in various shell payloads, (mine seemed to like the 3" Winchester High Velocity #5's in 1 3/4 ounce), on large pattern board paper. If I wasn't against spending a little more for shells, (which I'm not), I'd try some Remington Hevi-shot. Personally, after what I've seen first hand, I won't shoot anything else at a turkey. It just shoots too darn good when compared to lead. That's the reason I suggested you buy a choke tube rated for Hevi-shot... just in case you lead didn't pattern well.

A side benefit to Hevi-shot is in retained downrange energy. Because of its weight, a #6 Hevi-shot pellet hits as hard or harder than a #5 lead pellet. So, with a 3" load of #6 Hevi-shot, you increased your pellet count on average, 75 - 100 pellets, while getting the same killing power as a #5 lead sized pellet. That's pretty dang nice. You can read a bit about it at this link... along with what choke makers are rated to handle Hevi-shot. http://www.hevishot.com/faq.html

If I stilled had my 870, I'd shoot Hevi #6 with a .665 or .670 rated choke... good look and hunt safe.