Now, you want to find not only where the piece goes, but how it is oriented. In order to do this, you pick a standard color for each cubie. You then trace only the sticker of that color -try moving your index finger along its locations. From where the sticker lands, you can then "reconstruct" the cubie at any state. I use standard colors to absolutely avoid confusion, normally the top/bottom colors for the U and D layer pieces, and F/B color for E edges. In my examples, it's yellow/white, and red/orange if those colors are not on the cubie.

Before you start tracing anything, make sure that you know the cross very well. Even though it does something, remember it as an arbitrary sequence that operates on the cube in the beginning. Memorize it as a set of five moves. Number them, or even memorize their notation to make them very clear.

For now, I'll take only scramble #1:

Consider the FR slot corner: It's YGR.

Look at the scramble and find YGR; in this case it's at FLU

Now, focus only on tracing the principal sticker.

Apply the cross moves, following the sticker. I find that helps to have the tactile reinforcement of following through the location of the sticker with you index finger.

You should now know that the sticker ended up on D[LF], from which you can reconstruct the entire piece.
Note two interesting things:

• The piece was only affected by one cross move. This will happen often; sometimes a piece will be moved by all the cross moves, occasionally by none, normally by a few. 40% of the pieces (8/20) areturned every move, so you should expect each piece to be moved during about half the cross turns.
• The piece ended up at its "home" location and orientation. This can be very good because it leads to easier algs (normally). In this solve, though, it actually doesn't end up being useful..

Now, do the same thing with the FR slot edge: RG. (It's at DB.) Its principal color is red, so follow it and you'll find that it lands at U[R].

So for the F2L slot, I've traced two stickers, and I remember them with those stickers' positions (it helps to associate them relatively if you'll have to remember the for a while- for example, here they are both facing up/down).

This corresponds to the F2L arrangement to the right. Here you could just decide to do U' F U' F' U' L' U L and group the F2L. However, since you will have to trace the other F2Ls through this point in the solution, you might as well see if any of them are nicer. Keep this slot in your mind, though, if you can.

If you trace the other four F2L through the cross, you will find that they are in the states below. (Left to right: FL, BL, BR, FR)

After tracing all the pairs, I have something in my mind that corresponds to the cube on the right (I'm using color on it to distinguish order: white -> yellow -> orange -> red). I remember them as four ordered pairs of stickers, like "these two, then these two, then these two, then those two." I always remember them in the order of FL, BL, BR, and FR.
Sometimes, I won't actually reach this step. If there is a really short F2L for a slot that I'm almost sure I will use (like L'U'L), I'll just trace the remaining slots through those extra moves for continuity in tracing each sticker.
You don't actually have keep all of this in your head. You can trace each pair, and then see how easy it will be to place it. Then, you only have to keep in mind the most easy pair so far. Then, however, you have to retrace all the others, going through this state again.

During a solve, you'd never "see" this, because you don't trace LL pieces until the end, but the F2Ls are located in the total cube states on the right.

Now, you have to pick an F2L slot, and perform the placement moves in your head. At first, I suggest tracing the pieces themselves through the moves you memorized, to be certain that you haven't made a mistake (actually, on you very first solve, you might want to trace from the beginning every time).
In this case, there was no really nice one, so I chose BL and did U' R U2' R' U' B' U2' B. Notice that I could have saved a move and done U' R U2' R' B' U B. However, I was doing a little look-ahead, and decided on this move for an easier FR slot.
Good lesson: I did not notice that U' R U2' R' B' U B would have turned the BR slot into (R B' R' B) or (U' R' U R), which would have been quite nice. However, since you don't have a comprehensive view of the whole cube, this will happen often without you knowing it. Maybe a certain path would have led you to an LL skip, but don't be disappointed for missing such an opportunity (especially if you are going for speed memorization).
If you are trying to go for fastest execution, though, you might want to try a few alternates to see a little further ahead -or you might try a few different slots. Or maybe you'll do something weird or insert an extra move (as in here) to get a better arrangement for the next F2L.
Try to think as creatively as possible.

Until now, you have constructed a partial solution through cross and one slot. When you go on to the rest of F2L, you can treat it like the cross, or an extended cross: a sequence of moves to apply to pieces. When you trace other pieces through the moves, try always to remember what the moves are doing, to avoid small mistakes (a "small" mistake of a wrong turn affects at least 8 cubies).
Note that for the the rest of F2L you only need to utilize the last placement moves, though you should be able to recall the beginning later (for LL, or to double-check).