Something Better than Elmer's Glue?

Aleene's Tacky Glue is great for what you need. Aleene's also makes some other craft glues. Also, Elmer's does make a tackier craft glue.

For really difficult projects, E6000 is the best, but it's super sticky and not to be used be a little one.

Another vote for Aleene's Tacky Glue. If you check the label, you'll see that it will handle fabric, wood, plastics, most anything, yet it is safe to use, nontoxic, dries clear, and sets up faster than basic Elmer's school glue. You can also use a fabric glue but try it yourself first -- some types of fabric glues are runnier than Tacky Glue and your daughter might end up squirting out a ton of it (and it does not come out if it dries on any clothing). Aleene's makes a wide range of glues so check them out -- any craft store chain should sell their glues. I would avoid rubber cement . See this, from the National Institutes of Health, and note the second sentence about young kids:
http://www.nlm.nih.gov/medlineplus/ency/article/002820.htm

Tip: Tacky Glue can be a bit thick so be sure to keep the nozzle at the top clear of dried glue, store it upside down as someone else noted (great idea) and for your daughter, consider squeezing the glue out onto a paper plate and letting her use a cheap, disposable kiddie paintbrush to apply glue instead of having her squeeze the bottle. The Tacky Glue bottle could be a bit frustrating for her to squeeze as it's not really ideal for very little kids' hands, and getting out the thicker-than-Elmer's glue can be tough for a young one. I used the paper-plate-and-paintbrush method a lot when my daughter was little and still use it myself sometimes when crafting!

I second the suggestion of Aleene's Tacky Glue. I always have a bottle upside down in a cup on the counter. Love the stuff, washes off while wet and you can put some in a dish with some q-tips for spreading or just have her dip the pieces into the glue and apply.

I should add that I used this with my groups of preschoolers without incident.

Aleene's Tacky glue is great. We have moved to that completely with our Sunday school kids because it works so well.

Our Elmers glitter glue works great on everything. Have fun!

Glue sticks are used a lot in school but the adhesive dries out eventually.
I wouldn't let any 3 yr old use any glue assisted or not.
Heck - even 5 yr olds will put paste and play dough into their mouths.
3 yr olds can (and do) shove stuff up their nose and into their ears.

ETA: Another thought, how about fabric glue? That might work well with things like yarn, poms and foam.

My first thought was rubber/paper cement. You can get it at most craft stores, and I use it quite a bit in my craft work and scrapbooking. It comes in a can, with or without a brush. You brush it on your material (or paper) and then firmly press your object down. It takes a little while to dry, but I like it.

Regarding Elmers glue, there is a stronger one called wood glue that is better than the standard school glue.

Have fun with your craft time with your daughter!

I use E-6000 from Hobby Lobby. It's clear and sort of runny and it works very very well. Like super glue but it's in a tube and you can use it on anything. It's smelly so I use it in a well ventilated area such as my dining room with a fan blowing the scent away.

It glues all my crafts very well.

Weldbond Glue is my favorite...non toxic, odor free, and water soluble but super strong. (I order it on amazon by the gallon. I'm a mosaic artist but use it for everything)

There is a glue for foam and it does work well but I have never used it for poms. There is also a glue that the name is totally hiding from me right now. it has the word Tacky in it. You can buy it at a craft store and it comes in a squeeze bottle. It does take a little longer to dry since it is thicker.

I always like the great big fat glue sticks. Not the smaller skinny ones. They work pretty good on arts and crafts.