1. Go to a rock gym and get a belay card. The skills are fun to learn and belayed climbing is typically easier on your joints than bouldering... Being able to balance comfortable on your toes while maneuvering on slabs is the numero uno climbing skill that transfers to scrambling.
2. Strengthen your ankles: practice balance on uneven or wobbly surfaces at home. (Try standing on one foot with your arms up in the air and your eyes closed, and you will engage the muscles that stabilize your ankle, thus building strength and balance).
Everyone else has already given great advice, and scrambling is really something you only "learn" by doing, so you can definitely start training on the easy scramble "trails" in the Gorge (Munra, ROA, the NW ridge of Table Mtn) and then see if you can tag along on more adventurous off trail trips with others.
In terms of physical skills (rather than navigation), I'd like to echo these two points Charley makes. As Jess noted, scrambling is easier if you already know how to rock climb, so getting some experience with actual climbing will teach you balance and leverage skills that will translate easily into 3rd class scrambling (ie, that requires your hands).
For rougher walking, strengthening your ankles is definitely important, but you'll also need to work on your balance for being able to cross loose talus, descending steep slopes, etc. Learning how to stay balanced in precarious or unstable terrain is crucial. Practice descending blocky talus (I recommend the talus field south of the summit of Mt. Defiance) and moving quickly downward (without poles!) by stepping from rock to rock without stopping. Also find some steep sandy slopes you can descend where boot skiing helps save your knees but requires you to maintain your balance as you basically run downhill.
The key to walking through loose and scrambly terrain is to learn how to keep yourself on the edge of losing your balance, and use that to maintain momentum (sort of like a controlled fall; think again of skiing). Combined with learning how to pull with your arms and maintain balance over your feet while climbing, you will have a good physical foundation for beginning to attempt more challenging routes.
Go do Whittier ASAP! It's super fun, and fall is the perfect season for it. For more of a drive, the Alpine Lakes is generally pretty rugged and some of the "trails" require a little more scrambly action than normal. If you ever find yourself walking through a trail-less field of granite, play the "lava game" by hopping from rock to rock - this reduces impact on the grasses and also builds strength and balance.