Finding a pattern is the hard part. There are several to choose from. I have a few in the pictures section at https://www.facebook.com/MakingPuppets, but the one above is my favorite as it has shoulders built-in.
You can either cut two of the above and glue them together or cut one big one with the right edge being a fold between them (this removes and seam line from the puppet's chest)
Spray glue the foam and the back of the fabric and stretch the fabric over the foam as in step 3 above.
Cut the seams as close as possible, even leaving a 1/8" gap between the pieces is fine. Then stitch them together using a whip stitch or the "Henson stitch" shown here:
Cut a pattern the shape you want for the hand and arm. Fold your fabric in half and cut out 2 sets (4 pieces total) and sew each set together
If you just want standard arms, stuff the fabric sleeve When packing the sleeve leave a little gap in the stuffing where the elbow wold be. This makes it bend in the right spot.
If you want the fingers and arms to be pose-able, take wire (16 gauge works well) and make 'fingers' to match your hand
Stuff the fingers with Poly-fill and tape to fold in place. Squeeze the fingers together like if you were touching all fingertips together to make it smaller. Now push it into the fabric arm you made, pulling each finger into its corresponding spot. Now fill in the extra space in the sleeve with more poly-fill.
When you get to the elbow, stop.
Cut the wire about 1/2" longer than the joint and fold the extra back on itself, making a small loop. Loop another piece of wire through these holes, bending it backwards, again another 1/2" longer than the length of the upper arm. Fold these ends as well so it doesn't poke itself through the body.
Fill with poly fill leaving it loose enough at the elbow joint you made so it can bend.
Attach the arm to the shoulder. You can sew it on as I did, glue it on or use doll joints. The choice is yours.
Attach the head to the neck, add clothes (close for 18 month children work well on mine) and have fun!