what you have to do next, depends on the situation. In best case you are at least a 2 ship. If you are allready on cap you should be flying in line abreast and be setup to air-to-air. If not, you should switch A/A and go line abreast. In the F4 it is better to fly slightly below the bogey because the radar is not very good in finding targets against a background ground clutter. You then turn to the bearing provided in the BRAA call. You want to be as fast as possible, without using burner, so set mill-power (but first maintain line abreast on the new heading). Next is to identify the bogey as bandit. Without specific ROE normally you need two indications for a bogey to be declared hostile. First would be Jester calling no IFF. The second could be a call from awacs, or a visual ID. That was the big problem during vietnams Operation Rolling Thunder. But in DCS you can aply other ROEs, so if you like engage as Jester calls him a bandit. When engaging you want to be faster, higher and full hot on the bandit. Meaning you put in full burner until you get supersonic and then make a smooth climb maintaing your mach at 1.0x.
You can tell jester to lock a specific bandit (helpfull if its a group of at least two bandits) by selecting it in the Jester Wheel or you can snap Jester Context Button short and he will lock the next available Bandit. If you have problems finding the bogey on radar you can manipulate the radar search by either using the Jester Wheel or switch to boresight mode by pressing the cage button. Jester focuses the radar emissions straigth ahead whichs means you have a nearly instantanious refresh rate but also lose SA on everything whats not in your boresight.
After getting a solid lock you have to wait until in range and also wait for jester to count the potatos. When you get your shoot indication you hold the trigger until the sparrow fires. After that you perform a cranking manouver and put the bandit on the edge of your radar beam and guide your missile in. Sending some countermessures is a good idea because the bandit will likely be firing at you as well. Given that you are not facing an enemy with Fox 3 missiles you should be in an advantage because you are faster and higher than your enemy, so your sparrow should reach him first.
Hope that helps.
As far as I know, the rudders are not designed to change course of the aircraft. As I understand there are only two usecases for the rudders. First to adjust the noseheading on final approach in the case of crosswinds to hold the glidepath and second you can use the rudders on high angles of attac and low speed to roll the aircraft, because aileron inputs cause the opposite reaction than intended under these conditions. And of course you can use the rudders to make smoother turns while banking like you would do on a light prop plane but honestly I didn´t noticed any positiv effect on my turns, so I just keep my feet off the pedals during the normal flight.
OmasRachE's post in F-4E Intercept Advice – Best Practice After a GCI BOGEY Call (AI Jester) was marked as the answer