Hi Gino,
Some great advice from Pruninggal. Here's some sites that explain tree root growth and mulching.
http://www.mortonarb.org/research/treeroots.html
http://edis.ifas.ufl.edu/WO017
http://www.treesaregood.com/treecare/mulching.asp
With that said, you can plant under your tree, but you will need to be careful and use plant material that can tolerate dry shade. Here's a site where you can search for those conditions based on your hardiness zone and several other requirements.
http://bestplants.chicago-botanic.org/query_simple.htm
Here's some other helpful sites. You will need to check the hardiness zones on these. This first site has a few plants that I don't recommend because they are very invasive. I never recommend ivy. If you aren't sure you can either ask here or search for more info at www.google.com
http://www.papillonsartpalace.com/plants.htm
http://www.pfaf.org/leaflets/grdcover.php
Some of my favorites are Epimedium, Ceratostigma plumbaginoides, Heuchera (come with many different leaf colors), Tiarella, Arum italicum, Campanula latifolia, hardy geraniums (a link to follow) & Corydalis lutea to name a few. Under vines I DO NOT recommend Lonicera japonica - Japanese or Hall's honeysuckle. It's an invasive pest.
Hardy geranium info.
http://geraniaceae.com/geraniums/
I suspect you are in zone 5, but if you aren't sure of your hardiness zone, here's a zip code zone finder.
http://www.garden.org/zipzone/
Newt
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.