Are there types of nomadic plants? Not plants whose seeds simply
travel great distances, but plants that actually travel? Is it true
that Mangrove forests can uproot themselves and travel great
distances?

Although mangrove trees' roots look like legs or stilts, it is
actually another tropical tree, the "walking tree" (also called the
"walking palm" or "stilt palm") that has an ability unique among
trees: it is capable of moving itself from one place to another. Red
mangroves and black mangroves are sometimes referred to as "walking
trees" because of the leg-like appearance of their roots, but the true
walking trees are not mangroves, but palms called "Socratea."

"The walking palm is a tree that forms stilt-like roots and, over a
period of time, picks itself up on them 'walking' out from under
fallen trees or toward light. This is the only plant that plants
itself wherever it wants to!"

Blackridge Primary School: The Rainforest
http://homepages.which.net/~matthew.shanks/page4.html

"The Walking Tree of the Amazon Jungle has the unusual ability to
'walk' on aerial roots, which serve as 'legs.' Inspired by a need for
more light, the tree lets roots on one side die off and grows new ones
in the direction it wants to go. In this manner, the Walking Tree
moves up to seven feet a year."

Walking Tree Press
http://www.walkingtree.com/walkingtree.html

"The poor quality of the soil combined with the struggle for the
sunlight have created some unique adaptations amongst the plants/trees
in the jungle, the best example of which is the tree known as the
walking palm. The tree is capable of new sending roots towards regions
rich in nutrients and severing the old ones, thereby actually
'walking' to the food rich locations."

Travel Notes - Peru : The Amazon Rain Forest 
http://www.travel-notes.org/amazon.html

"The walking palm is a tree that actually appears to be walking in
search of sunlight. It shoots out roots in the direction of the
precious light. As it does this, old shoots die such that the tree
actually changes its location. The base of the tree looks like a huge
cage with the roots serving as the base."

The Woodrow Wilson National Fellowship Foundation 
http://www.woodrow.org/teachers/esi/1999/costarica/projects/group1/Bunch/ 

"One particular plant we saw was the 'walking palm' (Socratea
exorriza)... Its name is given because it lacks a central root system,
and if during its growing stage, strong light comes from a specific
direction, the plant will move its top towards the light, more roots
will grow in that direction, and the roots on the opposite side will
gradually rot, giving it the advantage of a little lateral movement."

Lindblad Expeditions
http://www.expeditions.com/dersearch/derdetail.asp?action=ERD&expedition_id=&ID=6452

Here you'll find a photo of a walking palm:

Ronald Reagan High School: The Plants of Costa Rica 
http://www.neisd.net/reagan/departments/science/costarica2003/images/walkingpalm_small.jpg

Two more good photos:

Kan Family Web Site: The Costa Rican Rain Forest
http://www.kan.com/images/cr_rf/1326.jpg

Photo Album for PHP: Walking Tree
http://boonedocks.net/pa/images/0000000339_lg.jpg

Google search strategy:

Google Web Search: "walking tree OR palm" roots
://www.google.com/search?hl=en&ie=UTF-8&q=%22walking+tree+OR+palm%22+roots

I hope this is precisely the information you need. If anything is
unclear, please request clarification; I'll be glad to offer further
assistance before you rate my answer.

Best regards,
pinkfreud

The answer was very thorough and I appreciated the links.