To start using Nexmo, start off by signing up at https://dashboard.nexmo.com/sign-up
You start off with credit in your account that you can test out the apis with. As you start off with the demo, you are provided with an api_key, api_secret, and a number that is assigned to your account.
You can have 10 whitelisted numbers you are allowed to use in the "to" fields (depending on which api you use).
From here, you just have to make an HTTP call to start sending your first text. You can test out the API by making the call in your application or using a site like: https://www.hurl.it/ and entering the base URL ( https://rest.nexmo.com/sms/json? ) and entering the required parameters' name and value.
Full disclosure, I work at Nexmo