Just the fact that you know they will grow out of it in six months to a year will motivate you to find quality used or bargain basement items. Smart moms will clip coupons, combine discount possibilities and be close to the door when the sale racks are 50% off or more. A good winter coat can and should be purchased at a discount, as well as shoes, boots, and at least one good dress up item per season. The kids can get by in jeans and sweaters at school, but once in a while they need to look spit polished. Since there is a good chance it will only last a season (or maybe two), best to find a great department store with regular clearance sales.
Shoes are one thing that cannot be compromised. Since everyone stomps their feet upon the pavement a little differently, a pre-owned shoe can only cause your growing child's foot difficulties. Best to invest a little and buy a quality shoe even if you end up cutting corners in other areas like T-shirts, pants, sweaters, hats, and so on.
Underwear, socks, tights and long underwear need to be purchased. There are sometimes great buy two, get one free sales to stock up with. If you know your kid loves blue and will be a size ten next year, go ahead and buy it now. The worst case scenario is that you will have a birthday item on hand for one of their friends' birthdays. Most sales are off-seasonal in nature so if you can estimate that your kid will be basically a little taller and wider next season, it pays to shop ahead.
Consider organizing a hand me down system with neighborhood moms. The older child passes his clothes to the younger child, and you return the favor. Many times this is an ideal solution. Nothing gets thrown away, and money is saved. The chances are, they have so many clothes that they haven't come close to wearing them out yet.
Basic baby equipment includes a stroller, toys and didactic toys geared to increased concentration and learning. These are all great to pick up used at Goodwill or a garage sale. You may need to oil the wheels and adjust the seat, washing the cloth backing in the washing machine. At least when they spill on it you won't feel nervous about the money you invested.
If you get used teddy bears and other stuffed animals, wash them in the dishwasher. They will be sterilized and ready to be nuzzled against with no risk of bacteria or infection passed along from the first child.
Kids' things are transitionary to the maximum. Like a revolving door, clothes and shoes come in, go out, and the cycle only starts slowing down when they hit puberty. Even then, the same rules apply. Invest in shoes, get coats and jeans on discount, and use hand me down's whenever possible. In most cases they have hardly been used and can carry you through a season of everyday wardrobe needs.