Welcome to the forum!
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I think you're a little confused. The blockchain is the record of all Bitcoin transactions, and there is only one. You've finished downloading it -- your computer knows about all the transactions, so your Bitcoin balance is up-to-date.
As for the fraction of BTC you received, I assume you got it free from a "faucet" or something? From what you've described, it's unlikely that you earned BTC just by running the program. Did you paste a Bitcoin address somewhere?
As for getting more BTC, yes, you may want to consider joining a mining pool. In order to make any progress in mining though, you will need some nice computer hardware, not just a simple computer. You may have a graphics card that is capable of mining at a decent rate. Check out
https://en.bitcoin.it/wiki/Mining_hardware_comparison for more information on mining hardware.
Also see
https://en.bitcoin.it/wiki/FAQ if you haven't already, for more answers to common questions. That wiki site as a whole may be helpful to you as well.
Read around the forum and the wiki, and let us know if you have questions!