Sound design by itself is not different from composing music, it just works on a different level and operates on a different set of processes, rules and theory. Generally it may also focus on different types of sounds ("musical sounds" vs. basically any type of sound that fits for the purpose). But in a musical context or as well in a media context those two are interchangeable, i.e. there's not just (or at least I think there shouldn't be) "music composition" or "sound design", but rather just an entity, which comprises of elements of both and involves both ways of thinking. Sound is music and music is sound, IMO.
Because the two are interchangeable, the term "sound design" also means very different things depending on in which context you're viewing it. It might be sound design of a single sound element or sound design in the context of the entire soundtrack. And sound design is practiced on all levels. You might work on sound design of a single sound or you might work on an entire piece of sound design (which involves dialogue/music/SFX).
Basically you don't need some "special tools" to begin with, because sound design technically is just editing, processing (when needed), mixing and placing sounds on a timeline (in linear media). The tools are just sound shapers and the majority of sounds start from recorded sources, either self-recorded or sourced from sound libraries, rather than coming out of a plug-in.
I suggest reading a book or two about the aesthetics of "sound design" (in the context of an entire soundtrack, e.g. "Sound Design" by D. Sonnenschein [regarding film sound], "From the Shadows of Film Sound" by R. Bridgett [regarding games from the perspective of film sound] or maybe M. Chion's "Audio-Vision") to get a better idea of how it works in a visual context and what's its purpose. Or enrolling in a course or trying to get someone to mentor you regarding the topic (forums are pretty good for this). Then practicing on creating sound overlays for videos/games/something or actual projects and maybe trying out different approaches to see how they affect how the visuals and/or story are perceived. Then it's pretty much learning by doing basically.
Finding/creating specific or suitable sounds can be difficult sometimes, but there's nothing magical in the process itself (apart from maybe highly "inventive" sounds or inventive usage of sound). Offline or online sound libraries, preferably a recording setup to record things that you cannot or don't want to get from libraries, a DAW/audio editor with some plug-ins (FL Studio has a brilliant stock plug-in set) and a vision about what you need to create and off you'll go.