cheapest price per gigahash miner will probably be an antminer s1. They produce about $24/month of btc right now. If electricity cost isn't a concern and you want to try out mining that's a great unit to start with. If you try it and like mining, then I'd switch to something more powerful after playing around with that for a few months. If your electricity is free then buying the less efficient miners for really cheap will give you the quickest pay back.

In your scenario, I'd get a used antminer s1 to get my feet wet. Then I'd use the earnings to add an antminer S3 in 2 months. Then I'd use the combined earnings to switch out the S1 with an antminer S5 in 4 or 5 months. Using this methodology you'll be earning about ~$150/month in btc within the next 5 months assuming a 2% rise with each difficulty adjustment and no change in the value of btc. This methodology leaves you with an initial cash outlay ~$50 and would be your lowest risk entry scenario into bitcoin mining.

Alternatively if you want to risk more capital on the mining venture,

spondoolies ~$3500 unit gets you ~2.0btc per month at the current difficulty  (loudness may be a concern to you)

spondoolies ~$700 unit gets you ~0.54btc per month at the current difficulty (loudness may be a concern to you)

antminers  ~$420 unit gets you ~0.42btc per month at the current difficulty