How do I start a new way?

Hi,
since your question is somewhat unclear where your actual problem lies, I may point you to the Beginners Guide at the wiki. I think there you might find the information you need. Basically it's just about knowing some basic principles and picking an editor of your choice. JOSM and Potlatch are somewhat common I guess ...
If this is not the information you need, you need to be more precise in your question, maybe provide an example so that others can have a look and give you a better advice than pointing to the wiki.

This may help http://wiki.openstreetmap.org/wiki/Potlatch_2 and this http://vimeo.com/24984085

In JOSM, I would click on the junction node and hit G, this would unglue the roads and allow you to adjust the length and direction. To join the roads back after adjustments, make sure the roads to be joined are near each other and click J. Do play around and it would be great if you could go to cities like Berlin or Paris to see the dense roads draw by fellow users. This will allow you learn faster too.

Happy mapping !