Once you've finished the ribbing, place it on waste yarn, fold it and slip it over your dog's head to see how it fits.  It should be snug, but not constricting.  Here's how mine fits on Archie:
 
Put your knitting back on the needles, because here comes the fun part - deciding which Aran patterns you're going to use.  First, figure out how many stitches you have, then subtract two.  On your knitting, block off two stitches between stitch markers - I'll tell you what to do with these stitches in a minute, but they're not going to be part of your main Aran pattern, so don't count them right now when you're making your calculations.  For my sweater, since I cast on 72 stitches, this means that I'll be working with 70 stitches for the Aran portion. 

Aran sweaters are symmetrical.  For your dog sweater, choose a center cable pattern, and figure out how many stitches it takes.  You can find many Aran cable patterns on-line, and there are a ton of Aran stitch dictionaries that you can choose from.  I'm using the Compact Chevron cable as my main cable.  I find it easiest to follow cable patterns by reading a chart, but here are the written out instructions for the Compact Chevron cable if you want to use it too: 

Compact Chevron (26 stitches):
Row 1 (right side): Purl 1, knit 6.  Place next 3 stitches on cable needle and hold to back of your knitting.  Knit the next 3 stitches on your left needle, then knit the 3 stitches on your cable needle.  Place next 3 stitches on cable needle, and hold to the front of your knitting.  Knit the next 3 stitches on your left needle, then knit the 3 stitches on your cable needle.  Knit 6 stitches, purl 1. 
Rows 2, 4, and 6 (still RS, because you're knitting in the round right now): Purl 1, knit 24, purl 1. 
Row 3: Purl 1, knit 3.  Place next three stitches on cable needle and hold to back of your knitting.  Knit the next 3 stitches on your left needle, then knit the 3 stitches on your cable needle.  Knit 6 stitches from left needle.  Place next 3 stitches on cable needle, and hold to the front of your knitting.  Knit the next 3 stitches on your left needle, then knit the 3 stitches on your cable needle.  Knit 3 stitches, purl 1. 
Row 5: Purl 1.  Place next three stitches on cable needle and hold to the back of your knitting.  Knit the next three stitches on your left needle, then knit the 3 stitches on your cable needle.  Knit 12 stitches from left needle.  Place three stitches on cable needle, and hold to the front of your knitting.  Knit the next three stitches on your left needle, then knit the 3 stitches on your cable needle.  Purl 1. 

This pattern takes 26 stitches, so if you decide to use it, subtract 26 stitches from your total.  Since I started with 70 stitches, I now have 44 stitches to work with.  Remember that you're making your sweater symmetrical, and that you're placing your main cable pattern in the center.  So for me, what this means is that I only have to figure out what to do with 22 stitches on each side of the Compact Chevron cable.  I'm filling it up with Mini Honey Comb and the 9-stitch Twisted Rib Cable.  Here are the written out instructions for those patterns:

Mini Honey Comb (6 stitches):
Row 1 (RS): Purl 1.  Place next stitch on cable needle and hold in front of knitting.  Knit 1 stitch from left needle, then knit 1 stitch from cable needle.  Place next stitch on cable needle, and hold in back of knitting.  Knit 1 stitch from left needle, then knit 1 stitch from cable needle.  Purl 1. 
Rows 2 and 4 (still RS, because you're knitting in the round): Purl 1, knit 4, purl 1. 
Row 3: Purl 1.  Purl 1.  Place next stitch on cable needle and hold in back of knitting.  Knit 1 stitch from left needle, then knit 1 stitch from cable needle.  Place next stitch on cable needle, and hold in front of knitting.  Knit 1 stitch from left needle, then knit 1 stitch from cable needle.  Purl 1. 

Twisted Rib Cable - Left Leaning (9 stitches):
Rows 1, 2, 3, 4, 7 and 8 (RS): Purl 1, *knit 1 through back loop, purl 1.  Repeat from * 3 times. 
Row 5 (RS): Purl 1.  Place next 3 stitches on cable needle, and hold to front of knitting.  *Knit 1 stitch through the back loop, purl 1, repeat from * once.  Knit from cable needle as follows: knit 1 through back loop, purl 1, knit one through back loop.  Purl 1 from left needle. 

Twisted Rib Cable - Right Leaning (9 stitches):
Rows 1, 2, 3, 4, 7 and 8 (RS): Purl 1, *knit 1 through back loop, purl 1,  Repeat from * 3 times. 
Row 5 (RS): Purl 1.  Place next 4 stitches on cable needle, and hold to back of knitting.  From left needle, knit 1 stitch through back loop, purl 1, knit 1 stitch through back loop.  Knit from cable needle as follows: *knit 1 sittch through the back loop, purl 1, repeat from * once. 

For my sweater, I made up my 70 stitches as follows:

Stocking stitch for 1 stitch (worked by knitting and purling on alternate rounds)
Mini Honey Comb (6 stitches)
Twisted Rib Cable - Right Leaning (9 stitches)
Mini Honey Comb (6 stitches)
Compact Chevron (26 stitches)
Mini Honey Comb (6 stitches)
Twisted Rib Cable - Left Leaning (9 stitches)
Mini Honey Comb (6 stitches)
Stocking stitch for 1 stitch 
Total stitches: 1 + 6 + 9 + 6 + 26 + 6 + 9 + 6 + 1 = 70

The Mini Honey Comb makes a nice frame for the Twisted Rib Cable, and knits up into a lovely fabric.  I will post a photo of mine in a day or two.  You can use the same Aran patterns I'm using, or you can experiment and come up with your own combination - it's hard to go wrong.  Also, if the Aran patterns you want to use almost add up to the number of stitches for your sweater, or if they're slightly more than that number, you can fudge a bit by adjusting the purl stitches between the cables.  Nobody will notice if you have 1 or 3 purl stitches between each cable pattern.  If you're having trouble coming up with enough Aran patterns, you can also use the Mini Honey Comb to make up the difference by centering the Aran patterns you want to use, and flanking them by repeating the Mini Honey Comb to either side.  To illustrate for my sweater, if I didn't want to use the Twisted Rib Cables, I could fill out the rest of my 70 stitches by repeating the Mini Honey Comb 5 times to either side, and add one purl stitch at the ends. 

Mini Honey Comb, when you're using it over more than 4 stitches, is as follows:
Row 1 (RS): Purl 1.  *Place next stitch on cable needle and hold in front of knitting.  Knit 1 stitch from left needle, then knit 1 stitch from cable needle.  Place next stitch on cable needle, and hold in back of knitting.  Knit 1 stitch from left needle, then knit 1 stitch from cable needle.  Repeat from * for as many stitches as desired.  Purl 1. 
Rows 2 and 4 (still RS, because you're knitting in the round): Purl 1, knit the cabled stitches, purl 1. 
Row 3: Purl 1.  Purl 1.  *Place next stitch on cable needle and hold in back of knitting.  Knit 1 stitch from left needle, then knit 1 stitch from cable needle.  Place next stitch on cable needle, and hold in front of knitting.  Knit 1 stitch from left needle, then knit 1 stitch from cable needle.  Repeat from * for as many stitches as you did in Row 1.  Purl 1. 

Now, do you remember those two stitches?  Those stitches are going to make up the front of the sweater - the part that will go over your dog's chest.  On alternate rows, make one stitch at both ends of those two stitches.  So for your first row, you'll make 1, knit 2, make 1.  Your second row you'll have 4 stitches between the markers, so knit 4.  Your third row, make two more stitches - make 1, knit 4, make 1.  Your fourth row, knit 6.  To make one stitch, slip your right needle into a stitch the row below the stitch you're knitting from, and knit that stitch.  If you want to make it more interesting, start knitting the Mini Honey Comb pattern on even rows as soon as you have enough stitches to do so.  Continue to knit in the round until this section is as wide as the space between your dog's two front legs, then check back for Part 3 of this tutorial to find out what to do next.  Good luck!