Respect inevitably leads to trust. Do what you say and say what you mean. Under-promise and over-deliver can help manage expectations. If you are given a task you know will take you one hour, say you “should” have it done in two hours. You never know when you’ll get a phone call that eats into your time or when an emergency may pop up. If you get done in less than two hours, you will be perceived as a hero. If not, you can call and apologize that it will be “a little later” without much trouble because you said you should have it done. You didn’t promise that you would have it done. If people feel they can rely on you, they will trust you.
Also let people know that you are not asking them to do anything you would not do yourself, or have done in the past. Work hard and be seen working hard. If you come in early and see others who are there early as well, stop by and simply mention that fact positively. A simple word of recognition will go a long way to earning respect. Without respect, you will never have loyalty and without loyalty, you cannot trust your followers. Without mutual trust and respect, you cannot accomplish great things.
Remember: while your people need to be able to trust you, you need to build them up to the level where you can also trust them.