1)First, add dependency in project.json - "Microsoft.AspNetCore.Session": "1.1.0",

2)In startup.cs and add AddSession() and AddDistributedMemoryCache() lines to the ConfigureServices like this-

services.AddDistributedMemoryCache(); //This way ASP.NET Core will use a Memory Cache to store session variables
services.AddSession(options =>
        {
            options.IdleTimeout = TimeSpan.FromDays(1); // It depends on user requirements.
            options.CookieName = ".My.Session"; // Give a cookie name for session which will be visible in request payloads.
        });

3)Add the UseSession() call in Configure method of startup like this-

app.UseSession(); //make sure add this line before UseMvc()

4)In Controller, Session object can be used like this-

using Microsoft.AspNetCore.Http;

public class HomeController : Controller
{
    public IActionResult Index()
    { 
        HttpContext.Session.SetString("SessionVariable1", "Testing123");
        return View();
    }

    public IActionResult About()
    {
        ViewBag.Message = HttpContext.Session.GetString("SessionVariable1");

        return View();
    }
}

5. If you are using cors policy then sometimes it may give errors, after enabling
   session regarding headers about enabling AllowCredentials header and using
   WithOrigins header instead of AllowAllOrigins.