How do I show a basic search block on the front page that integrates with search api?
Search API module has the ability to integrate with views module. All you need to do is to install views module and then create Search view,expose search filter and in your views configuration choose Exposed form in block option to yes, now you will the see the search block from your search api and place it on front page, the block will take you automatically to the search page to show the search items.