Depending on what you are using this for will depend on what you can use to make it. What you would be doing is making a bot, which a lot of websites discourage. If this website gives you an account and their rules specifically state no bots, then you would have to be extremely careful or they will lock your account. In some cases this may even be illegal, in which case don't try it.
There may be some pre-existing programs out there to do what you want, which may be your best bet if you are not interested in learning any programming, however if you are looking to learn some programming then you will need to know a couple of things first.
First of all, does the webpage you are loading check the referring webpage? Many web forms and scripts do check the web page that you are coming from, especially if they want to discourage bots. The program that loads the webpage also tells the webserver what page it visited last. So to make a good bot your program should lie to the webserver and tell it that it came from another webpage on that site (depending on what's appropriate for the page that you're loading). If the webpage you're loading doesn't care what webpage you came from and it doesn't care if you are using a bot, then you don't need to worry about this. If they do care if you are using a bot then forgetting to do this is a surefire way to get your account locked.
Next, I would start learning about a program called "curl". You can sort of think about it like a commonly used command-line web browser. One thing that makes curl so powerful is you can specify almost anything your web browser would, including passing site cookies, webpage referrals, form data, and telling the webserver that it's just an ordinary web browser (like the latest version of Internet Explorer or Firefox). I will warn you that the curl documentation is written for those with a fair amount of computer knowledge, but it's an extremely powerful utility if you want to make a really good bot: http://curl.haxx.se/docs/manpage.html
If you get confused by some of the curl documentation, you may need to go back to some of the basics. Some things you might want to know:
- What is User Agent spoofing? -- Check out the wikipedia article: http://en.wikipedia.org/wiki/User_Agent
- What is a Referer and why is it spelled wrong? -- It's spelled wrong in the standard and now everything has to spell it that way. Check out the wikipedia page for more info: http://en.wikipedia.org/wiki/Referer
- Do I really have to use curl to make this work? -- No! It's just a very powerful way to do it that should cover your every need. There may be simpler solutions out there depending on what your needs are.
Finally, you need to ask yourself a few questions.
- When you click on the button, is it loading a webpage or doing something else on the page, like something with javascript? I.e., is it doing something server-side (loading a webpage after you click the button) or is it doing something client-side (doing something on your web browser only, playing a flash game, etc). If it's doing something client side you may want to consider an entirely different kind of bot such as one programmed using the Java Robot class. If it's not client side then I wouldn't bother with this.
- Once you load the first page, do you need to have a program actually read what's on the page, or just push the button? If you don't need to read anything on the page and the button link is always the same, you can save yourself a lot of trouble and just have curl load the url connected to the button. If you need to have the program actually read what's on the webpage then you will need to use an actual programming language. PHP is a relatively easy one to use because it has curl built into it, but you should write whatever language you are most comfortable with. If you don't know any programming then you may need to learn some for this to work.
The short answer is, if you are making your own program, you will need to learn some basic programming and basic information about how web servers and web browsers communicate first. Some of the things I've listed above will help you towards that goal. If you aren't prepared to do this then you should look for a premade program that might best fit your needs.