HS125 Captain reported an FMS failure during approach resulting in a course and altitude deviation. The Captain executed a go-around; reset the FMS and was re-vectored for another approach to a safe landing.

Narrative

We were being vectored for the RNAV XX approach. We had the box (FMS) set up with a fix and extended runway centerline. It became apparent that we were going to intercept inside that fix so I went head down and resequenced the box for the waypoint in front of us. I was the Pilot Not Flying. The airplane was on AutoPilot.We were assigned 1700 MSL and cleared for the approach. We had just gone from Flaps zero to Flaps 15 as the Glide Slope became alive and began to center up. We were at 160 knots. Suddenly the glide slope portion on the NAV display just disappeared. The needle; the scale; all off it; and the airplane began a fairly significant LEFT turn away from the airport. I was unaware if the AutoPilot had tripped off or what was going on. There is no 'AP tripped-off' horn on this jet. There were a few seconds of 'What the hell just happened?' as we flew away from the course. All indications were NORMAL on the flight displays except for the GS. IOW (In other words); there were no FLAGS.What follows was HIGHLY compressed... The FO said something along the lines of 'where is it going?' To this I said; 'I don't know; but fix it'. What I meant was; 'Grab the controls and put the jet back on the Localizer'. We were in a very slight descent and now about 30-40 off the proper heading. The airplane was slowing and now we were below 150 Knots. As a result of our speed the gear warning horn came on. This horn confused my FO who wondered if it had something to do with the Glide Slope anomaly. At this point; the tower chimed in with a 'Uh...where in the hell are you going? (My words; he was polite; 'You're off course; say intentions or something like that). I was super frustrated. I told the FO that it was the gear horn and told the tower that our plane had just [given] us a curve and that we were going around (GA). I then instructed my FO to GA. We went around because I didn't want to continue play stupid games in IMC weather below 2;000 feet while troubleshooting whatever had just gone haywire with the jet.Again; ALL of this happened in just a very few seconds.My NORMAL SOP is to lower the gear at Glide Slope intercept and then set the Missed Approach Altitude in the window. We had previously briefed the Missed Approach; but had just SKIMMED over it since the reported weather was 500 Over and Ten Miles VIS. Who's going to fly a MAP with 500 and 10 miles? YOU ARE SUCKER! As a result; I thought the MAP alt was 3000 when it in fact was 2000. [expletive]! We started a GA as we turned back toward the course and in a NANO second we were at 3000 feet. The confusion about WHY the plane had LOST the GS and WHY it was veering away from the centerline completely overwhelmed us. It shouldn't have; but it did. My expectation bias allowed me to sit back and think....'This will be a piece of cake; it's 500 and TEN!'ATC said; don't worry; maintain 3K but he sounded pissed. I don't blame him. I then told him what had happened and asked him to re-vector us for the APP. He did. This time everything worked as advertised and we landed without incident.I called the tower to explain what had happened. He didn't seem upset; but ZZZ1 is right next to ZZZ and we popped up like an unwelcome party guest right into their airspace.I've been doing this XX years. Just retired from the airlines and am pretty much amazed at how quickly things turned sour and disappointed with my performance throughout ALL of it.My FO is 'rough'. This NOT an excuse; but I was apoplectic when he just sat there as we flew away from both the inbound course and Glide Path. I consider him '[inexperienced]' with the way he handles the jet. 'If the FD tells me to do something; then it's GOT to be right; right?' NO!!! You don't follow the Magenta Line to the point of impact. He and a LOT of the new guys I fly with are looking for that ONE MAGIC BUTTON to push when things go wrong instead of grabbing the yoke and throttlesand turning the AP and/or FD OFF. I see it over and over in both of my jobs. As a result; I am probably not the best mentor; particularly when things go wrong. I try like hell to encourage hand flying and turning the FD OFF when the workload is low.When I try and offer advice; a lot of young guys don't want to hear it. Particularly those who have a lot more time in the jet than I do. So; I get frustrated when I see stuff unfold.To sum it up; We lost guidance and automation at a very bad time.We QUICKLY got off course below 2000 ft. We then got huge distractor; the gear horn.There was confusion about the FMS and FD status.The corrective actions were applied MUCH too late and improperly.He thought the gear horn was some type of warning about the FMS and the loss of Glide Slope.I was frustrated that he just allowed the jet to wander off course. It sucked across the board and I am embarrassed by the way we handled it but more importantly; the way I handled it. This is an easy job. Till its not.