Basic beginner slalom advice
1. I completely agree with Yuki about watching and imitating the strongest skiers' starts, skating and poling: (A) whatever speed you have at the top of the course carries through the whole course (for physics junkies, google brachistochrone curve and consider the ski racing implications of that late-1600s cutting-edge mathematics research); (B) it allows you to use more inclination for the first few turns, skiing cleaner.
2. Think about nailing the first three turns WITHOUT worrying about getting close to the gates. You want to ski cleanly and fast from the ground up, not sacrificing the turn in favor of getting your boots a foot closer to the pole. (A) you can actually see the first 3 gates from the start, so you can plan this ahead, including your exit line from the third gate; (B) it's one of the few doable goals--a lot of the rest of slalom, for beginners, is recovery; (C) speed at the top of the course carries through the rest of the course (see above); and (D) there is, actually, rhythm to slalom and if you ski those first three gates cleanly (instead of with that horrible scraping soundtrack as you stay on your heels or you reach across your mid-line to "lunge block") you're more likely to do well on the next 12 as well.
3. Get forward as much as possible at turn initiation. It may help to think about pulling your feet back, or pressuring your front boot cuffs, or mashing a bug with your ski tips, or pulling your elbows forward--find what works for you.
4. Don't worry about cross blocking. Just ski and let the gates come to you. But it's nice to have a helmet with a face bar. (Types the guy who got his face bar only after going through a pair of glasses and several bloody noses.) BTW, don't use your helmet with the face bar for downhill racing or super-G, but those events may be a few years (and different hill) away for you, at present.
5. When not in the gates: Balance training, balance training, balance training. Slalom requires, at higher levels, impressive lateral and fore and aft balance skills. (And at lower levels, the way I hack in the slalom course, it, ahem, requires impressive recovery skills.) Rick has some nice on-snow feel and balance drills posted elsewhere hereabouts (and also posted over at www.modernskiracing.com
) Balance is also a progressive skill you can work on in dryland exercises appropriate for your current level of balance. The bosu and swiss balls are nice home training tools for dozens and dozens of different exercises.
Good luck and have fun.
(Just back from Summer break and ankle rehab. My projected first day back in slalom poles November 7, Copper Mountain. This year: New boots, rehabbed ligaments, no excuses...)