Different parts of the country have variable challenges for each kind of plum. Maybe you can tell us which area of the country the house is in order for any of us to start judging Japanese type plum recommendations for you growing them if we can. 

Most all of the top 10 posters on these forums have plums and have good answers for specific ways of building your own plum tree. We rarely have reasons to start new trees from seeds though (ouch), but the yard you speak that has a tree, are you sure it's plum. A good way to ID any species of tree is by the nature of the flower such as flower color and number of pistils etc. So another way to progress here (besides location data) is uploading any pictures. 

I would like to add a note about plums now, One of my top picks are always Japanese type plums of which I have several of. Lydecker is my favorite all time best plum here in the PDX area at least, but it takes very long for it to ramp into production here. The fruit quality may degrade if or when production ramps up so I can't say that will ever happen here being a plum bred by the breeder near to the colder state of Minnesota conditions. 

The front desk solutions during the volunteer years at exchanges of grafting wood advertised mason bee nests to improve pollination but this would actually not work in cases of the plums I have been in observation for in my yard because I think the weather is too cool during flowering right here. My observations are that what fruits set are through asexual means without pollen. For instance I'm 100% sure Nadia plum is (under my observations) considered asexual, meaning without bees. 

I'm learning this same cool weather trait with apples as well but that's not right here to talk about apples, right?