Selection of a rack depends on five things for me and is nearly always route dependent:
1. Type of route (rock, ice, snow, mixed)
2. Condition of route (dry, frozen, winter, etc)
3. My level of climbing skill (ability to run out protection, simul-climb or go unbelayed, etc.)
4. My level of familiarity with the route, gleaned from other climbers' beta or my own past experience on the route
5. My desire to minimize weight.
Often, for me, and especially, if Alan is along, I need to consider the fact that the last climb probably left my rack a little thin because we left most of the gear on the mountain rapping in a storm. Which means I enjoy the luxury of assembling a new rack frequently ;)
OK, here's what I'd start with as a bare minimum for the following route types:
1. Alpine Rock
Really Thin cracks. I still use pitons. KB's and Bugaboos. They hold better in thin cracks than tiny stoppers. You don't typically learn this until you've unzipped a line of thin stoppers. I use both BD and Camp pitons, and the latter are a little lighter and cheaper, and work fine. I use Ushba Alpine Ti pitons as well, although the titanium is not as malleable, so you have to be careful with placement - they aren't as forgiving, and require some experience at placing pitons.
Thin Cracks. I take a selection of stoppers, currently I climb mostly on ABC huevo wired stoppers, I like their shape and they are light and easy to place and easy to get off the rock if you take the time to learn the color coding. Sometimes I also take a few of the smallest size angle pitons as well.
Medium to Large Sized Cracks. Here I alternate between slung hexes (dyneema slung wild country hexes) and slung cams (DMM 4CU's). Hexes are lighter, cams are easier to place. Often I'll forgo cams for tricams, which are lighter at the smaller sizes.
2. Alpine Snow and Ice
mostly depends on route condition. early season and pickets are the go-to-pro. I use Yates expedition pickets (10 oz). I almost never use flukes, but have wished for them in some cases of very soft snow, when I've instead used things like my snow shovel, helmet, backpack, etc.
As the spring snow melts and the ice goes white (true alpine ice, the kind you find in gullies) Ushba Ti screws work great. I wouldn't use Ti screws from Camp, Salewa, or the cheapos at REI - they are tough to screw in even white ice. Glacier ice - blue ice - Ti screws are less effective, and this is where you need to begin making the switch to steel screws. Spend the weight on the express handles (I use BD) - it's worth the time gained making placement easier. For late season black ice, or spring/winter/fall melt-freeze water ice, steel screws are really your only option. The best three types of steel I have used (in terms of placement speed) are BD Turbos, Grivels, and DMMs, in that order.
Spectre Pitons from Black Diamond - everyone needs one of these. They are infinitely useful when you start to climb alpine melt/freeze routes that have water ice on them. Ushba makes a Ti version and it works great as well. For rappelling ice routes, an abalakov threader and a 22cm screw is invaluable for making V-thread anchors.
Only one option, anymore, worth bothering for: the ultralight 8mm Mammut Contact slings. If you are climbing water ice or rotten rock routes consider taking two Yates screamers to protect belays from shock loading.
Right now the ultralight standard is defined by the Trango Superfly - 30 g. Wait til spring and you can get Camp Nanowires for 28g. For locking, the lightest out there are the ABC Electrowires - 47g.
OK, so what you need to start will depend on what your partner has, but for now, let's assume that this rack needs to get you up moderate grade Class 4-5.5 routes in the Cascades, Rockies, Sierras, where you aren't placing a TON of protection on each pitch and you may run into steep snow and rock, with the very occasional short section of water ice.
First, start with the package the Pro Mountain Sports has put together, they call it the "Ultralight Starter Rack", go here:
Ultralight Starter Rack at Pro Mountain Sports
Add a nut tool (Ushba Ti), 3 KB/Bugaboo pitons, 2 angle pitons, 1 17cm express screw, 8-12 60cm x 8mm Mammut Contact slings, 2-4 120cm x 8mm Mammut Contact Slings, a few dozen Trango Superfly Biners, 6 or 8 ABC Electrolites, and you should be well set for most moderate grade mixed summer alpine climbing routes.