At this point, you need to decide whether to stay at your current job or take the new job offer. That's all that's on the table right now. Nothing else. You can't consider any counter offers because they're not on the table right now. BTW, 's not unusual to ask the new employer a few days to think it over.
If you don't like the new offer, then stay at your current job.
If you do accept the new job, then tell the new employer you have to give your current employer two weeks notice. Give your current employer your two weeks notice (put it in writing). You don't owe to give your current employer any information, other than "I'm leaving in two weeks." You don't want your current employer calling the new employer and possibly jeopardizing your new position. Within that 2 weeks, your current employee may or may not decide to increase your salary to entice you stay. You probably shouldn't tell them your new salary. Once you tell your current company you are leaving, you should really follow through with it to the very end. Unless their counter offer is too good to pass up, you should leave once you've committed to it. Because if you do decide to stay at your current company, you may be labeled as "That guy who tried to jump ship, but stayed." That's a chance you take and some people *might* see your move as a sleazy move, a bluff to get more money. A trust may be broken. That's why most employment experts recommend that when you say you are leaving, you should leave.