Well the best way to get your website indexed in the search engine of Bing is to integrate its webmaster account and submit your website's sitemap.

My best advice is to try to get indexed in Google first and foremost and let the other engines come as they do.

Google commands the vast majority of search traffic so it makes sense to target their search ranking first.

See , your pages will easily crawls by Google.The rate is Google crawls 577 sites per minute and the same time bing crawls 1.Google is the giant.If the Google indexed your site, then don't worry about others , they are also depending Google.
And also to speed up the crawling process by other search engines , you can submit your site map to their engine using webmaster tools.

Index first your site in Google. It will give your site a great advantage once it is indexed in google.

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I do not bother with Bing and Yahoo. If you will be indexed by Google and build some backlinks they will find you.
Even few days of difference in indexing should not be a problem and less work to do