Are you looking for something online tutorial, or for someone to write up a tutorial here? If you're looking for the latter, I can probably give you a good start. I'll need (one of) my (three) slide rule(s) in front of me to make sure I don't mislabel the scales, and the like. I might even be able to find the manual that came with my grandfather's Ricoh slide rule. Most (if not all) of what's there should be applicable to Pickett slide rules, too.
[edit: adding mini-tutorial]
I'm basing this mini-tutorial on my Pickett Microline 140.
Multiplication is usually done with the C and D scales, though it can be done with the CF and DF scales and even the B and A scales. Slide the 1 on the C scale (or CF or B, respectively) next to the first factor on the D scale (or DF or A, respectively). Then put the hairline on the second factor on the C scale and read off the product on the D scale. So 2x3 should look something like:
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| ------------------------------|----------
C | 1------------2-----3-----4--5-|6--7--8-91
D 1------------2------3-----4--5--6--7--8-91-
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The 1 on the C scale is lined up with the 2 on the D scale, and the 3 on the C scale is lined up with the answer to 2x3 on the D scale. I don't think I'm going to be doing any more ascii pictures.
With division, you do the opposite as multiplication. Dividend / Divisor = Quotient, so you line the dividend on the D scale up with the divisor on the C scale and read off the quotient (lined up with the 1 on the C scale) on the D scale. Using the same picture, 6/3 = 2. Cake and pie, right? Piece of cake and easy as pie.
So you want to extract square roots, do you? Well, that's what the A and B scales are for. To find the square root of 5, line up the hairline with the (first) 5 on the A scale and read off the answer on the D scale, or you can use the B and C scales, respectively. If you want to find the square root of 50, line up the hairline to the second 5 on the A scale and read off the answer on the D scale. Sqrt 10 = 3.16... and not 1 followed by zeros. That's why you need two copies of every number on the A and B scales. You have to be very careful of whether the power of ten is even or odd.
Squares are the reverse direction of square roots. Go from D to A instead of A to D.
Cube roots use the K scale with the D scale (K to D). The process is just like square roots. Likewise for cubes. 3/2 powers and 2/3 powers are simple extensions using the K and A scales.
What are the CI and DI scales for? They are the reciprocals of the C and D scales, respectively. You can use that as an alternate method for division, for instance.
The S, ST, and T scales are for trig functions. If you want to know what sin(35 degrees) is, find where the second number on the S scale is 35 (the first number is for cosine) and drop the hairline down to the C scale. I get about 0.573 or 0.574 on my slide rule. Windows calculator says 0.57357643635104609610803191282616, fwiw.
Arcsin and arccos are done in the opposite direction. You are limited in that you cannot get, say arcsin 0.05, but at that point, you may as well use small angle approximations, anyway and just multiply (by the R on the C and D scales - it's right at 180/pi).
The ST scale is for sec and csc, much like S is for cos and sin. Everything should work the same way. Likewise, T is for cot and tan.
LL1, LL2, and LL3 are for e^x and ln x. There are two of each of these scales: LL1 has one marked -0.01 and one marked +0.01
(if you look on the left side, that is. The right side has -0.1 and +0.1... there's a good reason for that, but whatever. I'm not going into that.). LL2 has one marked -0.1 and one marked +0.1. LL3 has one marked -1.0 and one marked +1.0. If you want to find e^0.125, find 1.25 on the D scale and read off the answer on the LL2(+0.1) scale. The +0.1 refers to the fact that you multiply 1.25 by +0.1 to get 0.125.
If you want to know ln 0.98, find 0.98 on one of the LL scales (LL1(-0.01)) and read your answer off on the D scale. Then mentally multiply by -0.01 to get approximately -0.02020.
The only scale left to talk about, then, is the L scale. That gives you the common log of the C scale (and that's why everything is evenly spaced on the L scale). If you want to know log 57.5, move the hairline to 5.75 on the C scale and read off the L scale (don't forget to add the characteristic, 1, to the answer, since L only gives you the mantissa). The reverse process gives you 10^x.