..."Hybrid"? Examples: Tangelo, Pluot, Aprium Etc...

And also if one wanted to make their own fruit how would they go about doing it? (Possibly a citrus variety)

The hybrids are created from the pollination of one species with another. Then the variety is propagated by splicing the new scions onto older root stock plants.
(Amazingly, all navel oranges are descendants by grafting from one plant.)

As others have hinted, you don't get a new fruit (or flower or tree, etc.) by "splicing" or "grafting". You do it by mixing pollen from one to the stamin or another.

Imagine trying to get a new dog by splicing one dog's tail to another dog. Doesn't work that way

Grafting, however, is often used to marry a desirable (fruit or flower producing) stem to a strong, disease-resistant root.

typically cross pollination (fertilizing the flower of one plant with pollen from another) in a controlled environment.

You would have to isolate the flower to be fertilized before it blooms, remove the stamens/anthers before they mature to produce pollen to self pollinate, pollinate it with the desired cross species, assuming pollination was successful wait for it to mature and produce seeds. assuming the seeds are viable (can't crossbreed everything) grow them to see if you got the desired result. As noted by the previous response, sometimes the rootstock of the hybrid plant is not very good so they graft to one with a known good root system capable of supporting the new hybrid.
Have fun!