To make the lining, I started with: (I don't have one, nice neat picture of all my supplies, so forgive the multitude of pictures)
2 - 15 x 13 pieces of the striped fabric.
1 - 20 x 10" piece of the floral fabric (this is for a gathered, elasticized pocket)
1 - 15 x 10" piece of floral fabric (this if for a flat, sectioned pocket)
1 - 15" piece of 1/4" wide elastic (not shown)

Start by making your gathered pocket. Take your 20 x 10" piece of fabric, fold it in half, the long way, so it's 20 x 5" and press with an iron.
Since I am using 1/4" elastic, I sewed a line at 5/8" from the fold. This creates a casing in which to put my elastic.
Now, put the right sides of your fabric together and sew along the long edge, making a tube. Once you're done sewing, flip it back so that right sides are facing out and press with an iron. *You could easily reverse the order in which I did the last two steps. You could make the tube first and then make the small casing for the elastic. Either way, it works. This is what it looks like when you're done.
The next step involves your piece of elastic and two safety pins. Pin one of your safety pins to the elastic at one end. Make sure the pin and elastic are going the same direction. Take the 2nd safety pin and put it in the elastic, only this time make it perpendicular to the elastic. Make sure the safety pin that is going the long way is small enough to fit through your casing.
Take your fabric and start pushing that safety pin through the small casing. You inch the fabric along the safety pin and gather it as you go. By the time you get to the opposite end of your fabric, the pin that was perpendicular should catch and not slide through the fabric.
Very carefully, grab your elastic and match it up with the end of your fabric and stitch several times to hold in place. Remove the other safety pin and do the same at the other end. It should look something like this when you're done. It won't look quite so gathered, though. (See my "bloopers" at the end of this post)
Lay your pocket on top of your lining piece and pin in place at the top and sides. I placed my pocket 4 1/2" from the bottom of the lining piece.
For the bottom of the pocket piece, you will need to create gathers in order for the pocket to fit within the 15" of the lining piece. Making sure that the bottom left and right sides are lined up, figure out where you want your individual pockets to be and what size you want them. Run a stitch from the top of the pocket to the bottom. Continue to do this until all of your pockets are created. Now, to finish the bottom of your pocket piece, create gathers on either side of your pocket stitching and run a stitch along the bottom. This will be very bulky due to the gathers. Hopefully you can see my gathers in the picture.
I was making this gathered pocket late last night. My very tired brain couldn't take it, so I decided that the 2nd set of pockets on the other side of the lining would simply be flat. It starts the same as the gathered pocket, though. You fold the 15 x 10" piece of floral fabric in half the long way so it's 15 x 5" and press with an iron.
Now, instead of creating a tube by sewing the long end, I decided to simply sew the edges of the pocket to the lining piece and then flip it up to create the pockets. This is hard to explain since there are no right and wrong sides to the folded fabric.
Here's how I did it. I measured 4 1/2" up from the bottom of the lining piece and drew a straight line with my water soluble pen. I lined up the raw edges of the pocket piece (not the folded side) with the line that I'd drawn and pinned it in place. I then sewed a 1/4" stitch along the raw edge.
Flip your pocket piece up toward the top of your lining piece and press. Now you have clean edges without having to create a tube. Pin your pocket in place at the sides.
Figure out the spacing of your pockets and use your water soluble pen to draw lines. I made mine approximately 3 1/2-4" apart. Sew along these lines to create your pockets. Then sew along the sides and bottom to finish it off.
We're almost there. I now have two lining pieces. One with a set of gathered pockets and one with a set of flat pockets.
Take the two lining pieces and put them right sides together. Pin in place and stitch around the three sides, leaving the top open.
Clip your corners, but don't turn the lining right side out. Now you need to box the corners just like you did for the outer part of the bag.
Put your hands in each corner of the bag and then bring your hands together to create two triangles. Mark a line 2" across (see blue line) on each triangle. Take this to the sewing machine and sew a line following your blue line.
There! Now your corners are boxed and you can turn the lining right side out so you can view your handy-work.
Here is the gathered pocket...
Here is the flat pocket...
Now that you've viewed your handy-work, you're going to have to turn your lining piece back the way it was - with the wrong side facing out. This is because you are going to put it together with your outside piece of the bag. As you can see, I've put the lining inside the outer part of the bag, turned the raw edges inside and then pinned all the way around. Woo hoo, we're almost there. Now just the handles...
To make the bag handles I used two pieces of the striped material 5" x approx. 27 ". You can adjust the length to suit your tastes.
Take both strips to the iron and iron them in half making sure you have a strong crease. Use your center crease as a guideline.

Fold in both sides to meet the center and press. Like this...
Now take this piece and fold it in half, like so... Repeat theses steps with other handle. Take both handles to the sewing machine and sew together getting your stitching as close as possible to the edge.
Place your handles in between your lining and outer piece and pin in place. Be sure to make sure that both handles are equidistant from the side seams on both sides and that they fit comfortably inside one another at the top (where it would rest on your shoulder).
Now, simply sew around the top of the bag as close to the edge as possible. This secures the lining to the outside of the bag as well as encasing the handles. You can backstitch over the handles if you want to make sure they're secure. I also ran a 2nd line of stitching 1/4" from the first line. I felt this gave it a more finished look plus added extra security for the handles.
Voila! Finished diaper bag. (Pardon my messy work area) Doesn't it just scream Spring?