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Because the Monero blockchain is private and untraceable, you won't be able to lookup your Monero Public Address and confirm that the funds have arrived like you might with Bitcoin. This is good for privacy, but bad for convenience.
At this point you have many options. You can print the wallet on paper, save it as a PDF or text on a USB stick, burn it to CD/DVD, etc. Most likely you will want at least two or three copies, stored securely in different locations. If storing digitally, encrypt everything with a strong password. If storing on paper, do not show the wallet to anyone else who can memorize your 25 word key, or take a picture of the wallet without your permission. Sending someone a picture of the wallet is the same as giving away all of your funds.
An incomplete list of Exchanges that support Monero is on our [Merchants page]({{ site.baseurl }}/community/merchants/#exchanges).