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I have seen websites snd even apps where people are using a search bar along with the button saying either search or just a question mark and also there are many sites which use only search bar as they treat clicking enter as the default action after putting the query(figured out at the time of user testing for our app).
What is the best way and is there any research that says that you should go with a button or not?
Nielsen Norman has some research on Search, and go into some detail regarding using a magnifying glass or similar icon vs. a label
The Magnifying Glass icon in search design
Summary: People usually recognize that a magnifying-glass icon indicates a search tool, even when it has no label. Unfortunately, showing only the icon makes search more difficult to find.
They mention that one disadvantage of using the icon only:
The magnifying glass alone makes it much harder to locate the search. When used without an open-entry text field, the icon takes up less space. Visually, it’s less prominent and, therefore, less noticeable. We don’t recommend the icon-only pattern for desktop websites. Icon-only search makes sense on mobile devices, because there’s less screen space and fewer icons and labels in general. But on desktop, there’s more to look at, and thus, it’s easier for the stand-alone search icon to get lost in the crowd.
If the icon is visually prominent enough, most users did recognize the magnifying glass.
As for using a button or icon to click on:
Because many users still have the habit of clicking a Search button to submit the query, it’s essential that clicking the magnifying glass submit the query.
Their own site uses a clear label for search, and they are intentional about keeping the text label 'Search' and hitting Enter will trigger the query as well as using the button.
as for using Enter to start the query, you can use both the keystroke or the button as @Shreyas Tripathy has pointed out.
A lot of times, usability or user experience comes to down what the users are used to. This leads to a lot of practices (or best practices) to remain rigid.
Hence, keep the button in addition to key/gesture shortcuts
I think the only websites that (successfully) get away with subduing affordances in favour of branding and style, are creative agencies. Everyone else is dealing with a relatively varied target audience and often has conversion as the primary goal. If your goal is to show off your graphic design or development skills, by all means stray from the common paths. However if you want conversion, whether that's newsletters, sign-ups or phone calls, you'll need good usability. This requires patterns, and using buttons for actions (like search) is one of them. Don't force users to think about how they should interact with your website, or what clicking on something will do. In the famous words of Steve Krug: Don't make them think.
