I feel for you, I work in HR and so spend all day talking to people about how to be a good manager and actually I don't much like doing it myself 
, it's the fact of being responsible for other people like you say, trying to strike the balance of supporting them enough but also not micro-managing/interfering - plus I don't know about you but even though my team are great there is always, always some drama, this person is having domestic trouble and needs to be supported, that one wants a promotion and needs to be challenged/developed, this one's having trouble with X client and needs help to sort it, it never ends!
What helps me and what I advise the managers I work with, is to make sure you have the basics in place, by that I mean good structures and routines, monthly team meetings, weekly 'catch-ups', regular 1-1s - all this keeps good communication in place and reassures your team they can always talk to you and raise things with you plus should mean you catch any issues early. You wouldn't believe how many managers simply don't even bother to do the basics! I try and set aside about an hour a day purely for 'management stuff', answering queries from my team, checking in with them, helping them out with any problems, giving them feedback/praise/thanks, explaining tasks/projects I want them to do, as well as the more regular meetings I mention above - it sounds a lot and doesn't always 100% happen but I view a really large part of my job as being to manage, so protecting my team time is important to me.
Also, re the training, what specifically do they want/need to be trained on and what are you finding hard about doing that? When we think of/talk about training we tend to think about going to sit in a classroom and taking exams etc., but for the majority of work-related things the best way to learn is by doing - like you say that's how you learnt yourself. So if it's processes/procedures they need to be trained on, I wouldn't even try to get them all together in a formal group setting (far too hard to arrange), I'd get them one by one to sit with me or whoever does the task (virtually if need be) and first just talk through it, let them watch it being done and ask questions, then the next one they do themselves under your supervision, then they do it independently but with a quality control check when it's finished, then they fully do it themselves, then they 'train' the next person to do it until all the team is competent. You can also ask someone who is trained/aware of what needs to happen to write up a Standard Operating Procedure/guidance document on that task (doesn't have to be hugely elaborate, just bullet points or a flow chart or similar) which will make it easier for the next person to pick up and understand...
Finally if your team are happy and your manager's happy it sounds like you are doing a great job and need to cut yourself some slack! I know the frantic feeling you're never doing enough very well, but sometimes good enough is good enough, IFYSWIM 