You say they will all NEED things completed immediately but actually they won't - they might WANT them immediately, but that isn't the same thing at all.
With regards to prioritising:
First rule is to always look after the clients you've already got. It's a lot easier to keep existing clients than it is to find new ones, so if you're working on something for Client A but new Client B comes along, then by all means find time to fit in work for the new guys, but not at the expense of looking after "your regulars".
Be realistic about how long it will take you to do something, and always add a couple of days to give you some wriggle room; it's not unreasonable to need a week to do something properly, and if you are doing stuff to a high standard then people expect to have to wait for it. Always give a day by when you expect to have work ready - don't say "I'll have that done soon", say "I'll have that with you by Thursday", so that you both have the same understanding.
If a client rings you in a flap, help out so long as you can do so without making work for another client late.
The best time management advice I've ever had (from someone else who runs her own business) is not to work from an enormously long "to do" list, as it tends to put you off and nothing gets done. Instead, pick three things for each day and just have them on the list for that day - I don't know why it works but it does!
Work out what your most productive time of day is - some of us are best first thing, some later in the morning and so on. Use that time for work you need to get stuck into, and fit all the little bits and pieces around it. If it helps have a set time each day for doing all the bits that don't take five minutes, just to get them out of the way.
When you're busy don't be afraid to leave the phone to go to voicemail, and get into the habit of not answering emails immediately unless they really are urgent. It's very easy to be derailed by something that could quite happily wait 24 hours to be dealt with - don't let other people dictate your day. If you might lose business by not responding immediately, have a voicemail/automatic email response that says "I'm out of the office at the moment but I will phone you back within 24 hours (or whatever)", and then do exactly that, but at a time that suits you.
And finally, make time for running the business, it's always easy to lose track of filing, accounts and so on when you are busy looking after clients. And then everything gets in a mess and takes twice as long because you can't find what you need.
Best of luck!