Do you need to soften it because it's cheap, you haven't conditioned it in a while, or because it's old?
If it's cheap tack (relatively new and stiff from the beginning) I clean first then use as much pure neatsfoot oil on it as it will hold. Coat after coat until it quits soaking in. You'll need to apply most of it to the "rough" side of the leather to get it to soak in properly. Bend and flex the leather in all directions to help get the oil to penetrate deeply. Let it sit overnight then wipe off any excess.
If it's good tack that's just old or not well cared for, I clean it well then use Passier Lederbalsm. It's not cheap, about $15-17 a can, but a little goes a LONG way. I put it on with my bare hands, rubbing it in well, to all sides of the leather. Let it set for a couple hours, then reapply to any areas that have soaked it all in. I will then gently flex any stiff pieces of leather and reapply if the balm soaks in while flexing it. Let it set overnight, then wipe off any excess the next day.
To clean, I like Lexol's leather cleaner or a plain glycerine bar. Both soften up the leather a little by themselves and won't strip leather of it's color.
Clean/condition it every day you use it until broken in. Keep the leather well-oiled and ride with it oiled. This will help soften and break in the leather and mold it.
I'd say any leather products made for quality leather will work fine: saddle soap and generic leather cleaners from tack stores are the best, but I have found Lexol works really well.
Neatsfoot oil, but it will darken it so if you don't want that they make the neatsfoot oil that is non-darkening. If you use is, don't coat it, apply it using a sponge type paint brush in thin layers, will soak in easier.