When I was first starting out, Red Heart Soft and Caron Simply Soft were my go-to yarns.  As my color needs expanded, so did the list of yarns I work with.  It can be tricky to work with multiple brands and find the colors you need while making sure the gauges line up; I've learned the hard way that you can't always trust the numbers on the label!  These are the yarns I used most often, which provide a wide variety of color options and work well together.  Now if they would just stop discontinuing colors I use all the time...

In addition to buying directly from the brands below, many of these yarns can also be found at JoAnn, Michaels, or Herrschners. 

Red Heart Soft
Caron Simply Soft
Bernat Satin
Wool-Ease
Deborah Norville Everyday Solids
Paintbox Simply Aran

Most of the time I use plain black craft eyes on my dolls, but occassionally I'll use clear eyes with black pupils built in and hand paint them.  All of my dolls use 6mm eyes with safety backings (the plastic washers that snap on), but there are many other sizes and styles available.

Consumer Crafts is great for buying eyes in bulk.  The prices are low, but shipping can be steep, so I usually wait until they have a special deal on shipping.

JoAnn is one of the few stores I've found that has plain black eyes available in person, so I also shop there when I need smaller quantities in a hurry.

Glass Eyes Online has an amazing selection of eyes, both standard and beautifully hand painted, at great prices.  They also have an Etsy shop.

Many of my dolls feature embroidered details.  When I need an exact color match, I'll use 1 or 2 ply of yarn.  When I need a slightly different texture, or the right color is available, I prefer to use DMC pearl cotton or embroidery floss.

Good stuffing is important for keeping the shape of your amigurumi.  I like Poly-Fil Dry Packing Fiber Fill, which is geared specifically towards dolls and stuffed toys.  It packs tightly and holds its shape well.

A good fabric glue is important for attaching pieces and securing loose ends.  Fabri-Tac is a great all-purpose glue for a variety of surfaces.  It doesn't set immediately, so you have some flexibility, but it dries very securely. 

When I need tiny amounts of glue and a more precise application, I use a gel superglue  like Loctite.

Spray starch is an optional but useful finishing touch when you need to set hair, embroidery, or other small details.  Stiffen Quik is my favorite since the nozzle clogs much less than other brands I've tried.