1) Make small talk: Get to know your coworkers so it does not always seem all about work. The key to making appropriate office small talk is to not get too personal too soon. Keep it general at first and then gradually work your way up to discussing family and more personal issues when your coworker is also moving in that direction of discussions. Also keep small talk at work limited (don't go on and on) to respect everyone's tight deadlines.
2) Help each other out: Show them that you are willing and able to help them out when they are under a tight deadline. Not only will they reciprocate for you, but they will begin to trust and depend on you.
3) Ask them for their thoughts and opinions: This is a great way to show them that you really value and respect them. People like those who they feel like them. Show them that you value their advice and feedback.
4) Get away from the office gossip: Even if you have resolved not too gossip about your coworkers, you can be guilty by association. When workplace gossip starts, excuse yourself. This will show people that you will not be involved in the gossip and that they can trust you.
5) Do little things like get them holiday cards or birthday cards, congratulate them on their good work, and offer to grab them some lunch when you are going out.
6) Let others take the credit when they have done most of the work. If you are incorrectly given credit and praise, acknowledge the coworker who deserves the accolades.
7) Handle workplace conflict seamlessly: If you are not directly involved in the conflict, stay out of it, at least initially. If you decide you must get involved, plan out some negotiation strategies ahead of time and facilitate the resolution of the conflict. The key is to show that you care, but without becoming rash or worked up.
These tips, while effective, may not work in every workplace and with every person. When in doubt, ask. Ask your coworkers how you can best help them. Most importantly, be sure to follow up and keep your word by doing what you say you're going to do.