How do I do it? My phone now doesn't come up with any homepage for any browser.
What version of Windows Phone do you have 8.0 or 8.1? It's been many months since I used WP 8.0, so I'm a little rusty there.
As far as Windows Phone 8.1, IE11 doesn't really have the concept of a homepage. New tabs are just a blank page. If you try to visit the main Bing page you'll be taken to Cortana, unless you have your preferences set to render the desktop version of the page. Even if you are have your IE preferences set to render the mobile version of websites, you can visit the Bing subpages: rewards, maps, etc.
The phone's search feature is wired into Bing and you can set your lockscreen background to be the Bing image of the day.
The adress bar is a also a search engine input. So what you can do is setting your IE default search engine to Bing and write your researches in the adress bar when you open a new tab.
As for the Bing Search app what are the behaviors of your Search button? My Lumia does not have Cortana yet and I have got the following behaviors : Bing Search with a simple press and TellMe with a long press (it was on the Windows button before). Perhaps it is the same with Cortana? Cortana with one kind of press and Bing Search with another one, who knows? Tell me if I am right or wrong.