I have no magic answers, but because I care so much about nursing and nurse unions, I'll give a stab at a few answers. It won't be highly organized, but more a series of random thoughts.
I'm chief steward at my hospital, which is a position I took as soon as we organized about 7 years ago, so I have never actually been a unit steward as such. However, I now chair a counsel of stewards and work closely with unit stewards at my hospital.
I'm assuming - maybe erroneously, since every place is different, that you have some sort of structure - a stewards council or whatever and a staff person assigned to your facility.
Use them as resources. Ask lots of questions and don't be afraid to ask for help.
Quietly just go around and make sure all the nurses on your unit know you are the steward for your unit and that they should feel free to come to you with any problems they have.
Spend some time slowly and carefully reading the contract. If there are parts you don't understand, ask someone more experienced.
Make sure you have a thorough understanding of the concept of Weingarten rights. If it's not clear, talk to your staff person about it.
Do a little informal one-on-one education with your co-workers about it - impress on them how important union representation can be in a disciplinary situation.
Spread your work around. Your co-workers will feel more invested in the union if they share the load a bit. If you have 10 phone calls to make, get a co-worker to make 5 of them. Once they have proven their willingness to participate, it will be easier to ask them for a bigger task next time.
Unions are about every one banding together to get things done and it is both better strategy and better ideologically to share the work.
Take baby steps - I don't know how your union handles things, but we like our unit stewards to be able to handle the early phases of a grievance or disciplinary action on their own, but not right away - go to some meetings with someone more experienced, but as soon as you feel able start handling the easy ones on your own.
build an e-mail list - e-mail is a good way to communicate low intensity info - updates about minor things. But I've learned that when you have something really important, - like turning people out for a rally - it has to be person to person so....
At the same time, build a phone tree - let's say at a wild guess, you have 40 nurses on your unit, get 5-6 others to join you on the phone tree, so you can make 5-6 phone calls and each of them call 5 people and the message gets out quickly with no one person doing too much.
Look for opportunities to build power and to solve problems in ways that build power. You know that old proverb about "give a man a fish... etc."?
If a manager is up to something bad, and you or a more senior person in your union goes in and shows them the contract provision they are violating and makes them back off that way, you've solved the problem, but you haven't built power.
But if you organize the nurses to take direct action of some kind - to demand a meeting with the manager, or with the manager's boss and present a petition or something like that, then the nurses have learned about the power of group action and learned to work together and they have changed the power relationship on the unit. It may be more laborious in the short run, but it makes your union stronger in the long run.
That's all for now. Hope it helps.