Piecing by hand can take forever! Think of hexagons and diamonds
Piecing by machine is just straight seams, usually with 1/4 inch seam allowances. It's the accuracy of the cutting and the seam allowances which matters.
Start with a design which is only squares and rectangles - there are loads of them on the web.
When it comes to the quilting you do need space to lay it out flat, though a baby quilt isn't that big and is a good project to start with. Use lots of safety pins or hand basting to hold the layers together.
Guetermann SewAll or a good brand of cotton in size 50/3 for the piecing. If the colours of the quilt don't demand an obvious colour use a neutral - grey or beige - will usually blend in without showing. For the quilting cotton is my preference, size 50/3 if you don't want it to stand out, 40/3 or 30/3 if you do. Your machine won't take any thicker and remember to use a thicker needle if you use thicker thread.
If you prefer to hand quilt, cotton perle size 8 or size 12 is what you need.
The choice of wadding is another personal preference - a lot of people won't use polyester for babies, but I don't see why not as it's not going to be in direct contact with the baby, it is lighter and will be quicker to dry when the quilt is washed. 100% cotton or 80/20 cotton/poly is much nicer to work with.