I have found the best way to implement Scrum, is using Scrum.
Have a backlog of tasks you need to do to move from your existing processes to Scrum, break these down into a number of 2 week sprints, and implement gradualy over a couple months. this helps people to get tp grips with each process, without bombarding them with new tools.
Initially I would introduce a basic sprint planning meeting, daily standups, and sprint reviews, and keep doing the work using the old methods. Then bring in more methodologies as the sprint continue.
In particular Scrum suggests that each user story should be a vertical slice, with all aspects of the implementation done together to deliver business vale ASAP. Design, development, testing, infrastructure, integration... This can be very difficult estimate, and even harder to achieve. You will only really get this right when you have a rock solid, mixed disciplinery team, and very strong engineering practices. Start by bringing togther dev and unit testing if you haven't already, then bring more parts of the process into each task.
With Scrum, it tells you how to do things, not what to do. Look to XP if you want lots of hard and fast rules. Much of getting a really effective team is working out what works for you. Keep an eye on velocity and see what improves it.
Regarding tools, a white board is great.
BEWARE THE POST IT. These are great for reminders and notes on your desk, but one day you walk into the office and see your beautifully organised sprint as a pile of confetti on the floor. Even the extra strong post it notes dry out and lose their stick after about 2 weeks in a room with A/C. I learnt this lesson the hard way.
Use index cards, with drawing pins and a cork board.
Excel is perfect for working out your velocity and burndown metrics.
We only use tools with distributed teams. Then we use Acunote for it's simplicity. It is really just a virtual cork board.
Track time in your time tracking software. Track story points on your tasks. These are not the same. The recent snow in London and resulting transport chaos, dropped our velocity by 35%, and hence our ability to complete tasks, even though the team was doing more hours with a couple key individuals and clients working from home.