Many types of plants have official registries (known as an International Cultivar Registration Authority, or ICRA) where you can register a new hybrid that you create. For some genera, the corresponding authority requires that the new hybrid be different enough from existing ones to warrant registration.
Once you decide which genus (or family) of plants you want to work with, you can look at the existing hybrids available (i.e. Dahlias) and see if there is a gap where you can introduce something new.
For example, you could make hybrids with the aim of producing a new yellow Dahlia. There are already plenty of yellow Dahlias in cultivation, so this one would need to be different somehow - perhaps it flowers much earlier or later than the existing ones, or it has a much larger flower, or is taller/shorter, etc. You would then look for suitable parent plants - ones that have some of the traits you need - and start crossing them together to try to get seeds.
An alternative, which is what I do with my Irises, is just to make crosses to produce something for my garden. This means you don't need to worry if it is very similar to an existing cultivar; it might just be that you want to see what happens when you cross two plants, or you might want something similar to an existing cultivar that you cannot easily buy.
Keep in mind that sometimes a cross will only work one way, even if both plants are in flower at the same time. In other cases you might want to cross two that don't flower at the same time, so you will need to keep pollen from the first one and use it on the later-flowering plant. You can usually store pollen in the fridge for many different plants.
