For western or english glycerin soap is a good choice. Be careful about saddlesoaping the very light colored western show saddle. Most have a topical finish and where it wears off the leather will darken. For darker saddles you can also use Feibing's yellow paste which has conditioners. Be judicious with the oil. A good english saddle is tanned differently from a western saddle and usually requires just a periodic cleaning. People often mistake stiff leather as being dry or needing oil but it depends on what part of the hide was used. If you think it really is dry, go with cleaning it first. Then reappraise after a week as it takes about that long for the moisture to penetrate the leather. If still not happy then repeat the process. Of the hundreds of saddles I've cleaned, maybe half a dozen have been oiled.
I use Effax soap/conditioner on all of my leather. Its fantastic stuff!
As said above, avoid oil if you can. I'll occassionally give a brand new piece of leather tack a light oiling, but after than it's just saddle soap and conditioner.