I think you are asking about grafting (in the title?) Not a necessary process to take cuttings of most plants. In addition to cuttings, layering can also work very well. Both terms, grafting and layering, should have tons of general info available via a web search, as well as simply "propagation."

What did you want to propagate? Different methods are known to work more easily for some plants.

Cuttings can be relatively easy for a novice - grafting is much more of an acquired skill plus requires significant horticultural knowledge.

Each plant has a rather specific requirement as to the timing of the cuttings and the best methods for encouraging root development. There are some great texts on propagation you can purchase or just Google the plant name and propagation for specifics.

WELL THE MAIN THING RIGHT NOW IS SUCCULENTS BUT I'VE TRIED LANTANA, BALLOON FLOWERS, AND THINKING MAYBE DOGWOOD TREE. THANKS!

Succulents often propagate from a single leaf. Leaf or stem cuttings of succulents do best if allowed to callous (dry where they were cut) before propagating. This helps prevent bacterial infections. The cacti/succulent forum is very busy, if you have specific questions, or would just like to browse existing discussions.

Balloon flowers (assuming Platycodon) should make seeds one can grow.

Lantana grows extremely easily by putting cuttings in the ground (or a pot.)

No idea about dogwood tree, but somebody should have some advice for that one.

thanks so much!