Getting Into Things - Getting Close
A key thing to keep in mind as your relationship progresses is that it should always be COMFORTABLE. This should be someone you can tell all your thoughts to, that accepts you for YOU and isn't trying to change you. So as you go from just talking to hand-holding to hugging and more, these shouldn't be sudden, scary, threatening changes. They should be changes that you both are comfortable with and both enjoy.
Start with Casual Touch
Every person has 'personal space' around them that they are trained from birth to respect. If you went into a room and someone strange walked up to you and stood right against you, touching you, while they talked to you, you'd feel upset and uncomfortable. So the first stage in a relationship is to get through that personal space issue - feel comfortable in each other's space.
To do this, start getting closer, gradually. If you're sitting on a bench seat together, don't leave a giant gap! Sit next to the person. Don't mash into them, but be gently against them, that this is an OK and normal thing for you two to touch. Don't make a big deal out of it. The aim is to slowly get both of you used to the touch-is-ok idea.
Touch while Flirting
Touch is a key part of flirting! When you come by to say hello, touch the person on the arm or shoulder (or whatever's near) to get their attention and say hello. If you see an eyelash on their cheek, reach over and brush it away gently. Make touch an "ok" part of your relationship, to build along to future touching!
The Cold-Weather Gambit
This is a cliche now because it's so overused. But forget your jacket or sweater sometime. Any excuse to wear each other's jackets or to wrap your arms around each other to "keep warm" is a good one :) It's a perfectly reasonable request - you don't want to be cold! But the results are quite nice too!
Move to Hugging Goodbye
Hugs are non-threatening once you're already used to touching each other casually. So the next time you say goodbye for a long period of time - say a week vacation or even just a weekend - give a quick hug goodbye. Nothing long and serious. Remember, you're sort of "training your body" to think of these things as normal and comfortable. Just a friendly "I'll miss you." The more natural it gets to be, the better! Don't rush through any of these stages, savor each one, and how special it is. The more you enjoy each stage, the better they are!