For sport climbing, buy this stuff, in this order:

• Climbing shoes
• Harness
• Chalk bag
• Large HMS / Pear locking carabiner
• Belay/rappel device (unless using a munter, but you’ll need a rappel device eventually anyway)
• Rappel sling / daisy chain / 8 feet of webbing (webbing is the cheapest option)
• 7mm cord, 3-4 feet (for friction knot)
• 3 small locking carabiners
• sling (3 feet or so is fine)

At this point, you’ll have enough to clean a route with.  You’ll then want to get a rack:

• 60 meter dynamic rope, between 9.5 and 10mm diameter
• 10-12 quickdraws

Do not buy any part of a rack before purchasing a rope.