The deployment of trams is only a recent thing, and limited along two lines. The Red line serves lower income areas (can be unsafe), and the Green line upper income. The two lines don't connect, but will sometime later next year. If you find yourself located along one of these lines, as well as your work, then they are great, but it is kind of unlikely. They are laid out in a way to bring people too and from the city center mainly.
Dublin is mostly a city based on buses. You can really get anywhere in the city on the bus easily enough, though it could be a little better, since Dublin still employs a Ferris Wheel layout. This means you'll find yourself having to go into the city center, transferring there, and back out again to your destination. I find the service on time, safe and clean, consistently across the city. A bus is the only way to get to the airport, we have no rail connection (still....). If you Google Dublin Bus right now, you'll find a lot of people complaining about strikes, but this is only temporary, and happens every 5 years or so.
When it comes to cycling, people do, mainly to and from work. Ireland isn't really a country that is good for cycling, because we are prone to changeable weather, high winds at winter, lots and lots of rain, this is the same all year around. Also, the terrain isn't very level, so you'll find a great deal of strong inclines, so as I was saying before, people do cycle too and from work, but really if the terrain is OK, and they have the option to shower and change at work, it isn't a casual thing like in The Netherlands. Lastly, our cycling lanes, while somewhat plentiful, are kind of unsafe. They added lanes across Dublin during the 90's and 00's, as part of an initiative to get people cycling to work, but few are purpose built and merely a line on the road/footpath, 95% of the time you're sharing that lane with a bus.
Our train system, is under developed, there is a rapid transit system (DART) all along the bay, that for the most part will server upper class areas. It is safe, clean and on time, but it is unlikely you will live there. There are a number of commuter routes that go further to the West, North and South of Dublin. If you are on the North or South, you'll have a pretty easy commute. If you find yourself on a particular route out West, there are two different stations, Heuston and Connolly, Heuston will involve an additional tram or bus to get into the city center. There are plans in the pipeline to connect the two stations via underground....we'll see.
Finally, I just want to say that public transport is reliable, clean and safe in Dublin, but expensive. If you are commuting regularly you'll be spending somewhere around 120E per month.
To answer your last question, you cannot bring your bike (mountain/racing) onto either a tram or bus. Exception to this rule, are the collapsible bikes that you can carry with one hand, I think they are OK.
Hope this helps.