Hi WCG!
No, you do not need that extra-training.
However, you need to be aware that your initial character is mediocre in the greater scheme of things, so if you go right out to kill a giant or dragon, you will just add to those foes' kill lists.
Basically, I suggest that you first travel a little bit around the map, visit various towns/retreats/halls and collect information about the surrounding areas. Visit forests and plains where normal predators like cougars are reported (cougars are better than wolves because they don't come in packs) and first do a little bit of hunting. Besides giving your character some extra experience, it also helps you get the hang of the combat system. As with everything else in DF, adventure mode has a lot of interesting details to watch out for.
When you feel confident that you know what you are doing, go to a cave and try to hunt down some of the regular cave denizens (lizardmen, troglodytes, giant bats,...). If something tells you "To prepare to die!", run away as quickly as possible - that's the boss out to turn you into a pulp in no time.
Once you are confident about your ability to fight cave denizens and also packs of wolves without problems, go to a town surrounded by caves only featuring semimegabeasts (cyclops, giant, ettin). Get your first quest there, which will be to kill one of those semimegabeasts.
Well, this is how your career as a dragon slayer can begin.
Deathworks