You live in an area where the fall typically has a few weeks of rain and night time temperatures get gradually colder. You don't have to do anything now except watch out for any squirrels who may think you have buried some large nuts. A clay pot or some chicken wire will stop them from investigating.
In the spring watch for sprouts. At this time I would dig them up and pot them in a four inch pot. Water regularly and they should be ready to be planted, pot and all, in the ground next fall. Dig up the following spring, repot one size up and repeat.
After three years you should have a small sapling ranging from twelve to twenty four inches tall in a six to eight inch diameter pot.
Seedlings may not follow their parents in the type of fruit they could bear and you would do well to cull your collection of the less vigorous ones. Some plum varieties require other plums to pollinate.
A detailed article here is summarized below:
Planting
Be prepared to plant more than one type of plum tree because
many types require cross-pollination to produce fruit, although there
are some varieties that can produce fruit on their own. It is also
important to choose a type that will work with your location. There
are three categories of plum trees: European, Japanese, and Damson.
Hardy European types work in most regions across the U.S. The Japanese
types flourish where peach trees flourish. There are also American
hybrids that work well in regions where neither European or Japanese
types flourish. Plant plum trees in well-drained, moderately fertile
soil in full sun. Avoid planting in low areas where frost may settle,
as the frost will damage your trees. If possible, find sheltered
position, such as a south- or west-facing spot out of the wind. This
will help the plum tree set fruit. Space standard-size trees 20
to 25 feet apart. Space dwarf trees 15 to 20 feet apart.
Care
Thinning plum trees is important to prevent branches breaking under the weight
of the trees. If branches do break, prune them back into the undamaged
wood, ideally cutting back to a natural fork to avoid leaving stubs.
Be sure to water the young trees heavily every week during the first
growing season to help promote growth. Then, water regularly. It's
best to water the plant deeply at the soil line, then let the soil dry
out (though not completely), and then water again. Water your tree
well into mid-October to give it plenty of moisture through the winter
months. Do not fertilize young fruit trees unti they have set a crop.
Once established, fruit production requires regular fertilizing all
year long. If there’s good fruit set, fertilize with one pound calcium
nitrate per tree or 1½ lb. 10-10-10. Cut back the nitrogen in fall and
winter to avoid encouraging new growth in those seasons. In the fall,
rake away all debris and fallen trees.
Prune early spring or mid-summer to avoid infection. The best time for pruning is
usually spring for young trees and mid-summer for established ones. Do NOT
prune in the fall or winter injury or infection may occur. Consider a tree wrap
or guard around the lower trunk, especially for a young plum tree.
Keep an eye on the lower bark and branches for mice or rabbit injury;
if this could be a problem, you may need to install tree guards or
fence in young trees with chicken wire for the winter.
Pruning:
Japanese Plum If you have a Japanese type of tree, the best pruning
method is to create an open center shape. In the summer of the first
year, cut the vigorous shoots that form on the top of the tree by two
or three buds. After about a month, check the tree. As soon as you
have three wide-angled branches, spaced equally apart, cut back any
other branches so that these three are the main branches. In the early
summer of the second year, cut back the branches in the middle of the
tree to short stubs and prune any shoots developing below the three
main branches. After the third year, remove any shoots in the center
of the tree to keep its shape. Japanese types require heavy pruning to
help keep them in shape and to produce better fruit. It is also good
to thin out the fruit on these types of trees. You should space the
plums about 3 to 4 inches apart on each branch.
Pruning: European Plum
If you have a European type of tree, the best pruning method is to
create a central leader. This shape features a central trunk with
branches that spiral out every 5 to 8 inches, making sure that no
branch is directly above another. The training for such a system
begins in the early summer of the first year, during which time you
should remove any shoots that form within 18 inches of the ground. The
end result should resemble a Christmas tree. European types do not
require fruit thinning because they do not produce as much fruit as
Japanese types. However, the fruit on these types should be spaced
about 2 inches apart on each branch. To help control pests and
diseases, remember to prune your trees to keep them open. You can also
mulch around the trees in the spring to help control weeds, but be
sure to remove the mulch in the late fall so that no pests use it over
the winter. You can also lightly cultivate the soil around your trees
in late spring to eliminate any pests in the soil. Pests Plum trees
can suffer from silver leaf disease, honey fungus, bacterial canker,
pocket plum, plum aphids and plum moth.
Harvest/Storage
Harvest have the best flavor when left to ripen on the
tree. You can tell when plums are ripe by applying gentle pressure
with your fingers. If the skin of the fruit feels soft, then it is
ready to be picked. Plums should come off the tree easily when you're
using only a slight twist. Unfortunately, the fruit does not store for
long, so must be eaten or preserved. You can also pick the fruit when
they are still slightly firm and store them in a cool place to fully
ripen. The best place to store plums is in the refrigerator. The best
temperature for storing plums is about 31° - 32°F with relative
humidity around 90% - 95%; if kept at this temperature, plums may last
for 2 to 4 weeks. You can also store plums by making jams or jellies.
Plums can also be stored by freezing or drying them (dried plums are
prunes).
Recommended Varieties For a Japanese type, try the 'Satsuma',
which produces large, dark-red plums. This kind is good for eating
fresh and canning because of its sweet taste. For a European type, try
the 'Stanley'. This type is works great for the eastern regions and
some of the northwest regions of the U.S. It is self-fertile, meaning
no cross-pollination is necessary. It produces medium to large plums
and is great for cooking, canning, or eating fresh. American hybrid
trees, such as 'Alderman', 'Superior', and 'Underwood', also make good
choices. These trees combine the taste of the Japanese variety with
the hardiness of the European variety. These types work well for
regional extremes.
Edit:@ashes999 asks "Why re pot?" If you re pot and grow them on you can select the strongest seedlings and plant them where you want them when they are ready. Even if you plant them where you want them and do not intend to move them one accident with wildlife or a pruner will ruin all your hard work.
Just as a note seedlings are not guaranteed to come true to form. Size, fruit and vigor can all change dramatically if the parent was a hybrid.
