If You Must Use Flash
Generally speaking, my understanding is that Adobe no longer supports flash in Linux - but that the Google Chrome project has decided to integrate and support flash on their own. So Google Chrome would be the only browser that you can use at the moment out of the box, so to speak.
I have managed to get flash to work on other browsers from a recent install on an older 32 bit machine by using a flash-sse plugin (Shockwave Flash 11.1 r102). This build is made for chips that do not have SSE support. I don't know what machine you are using but this is important information.
This was not on an Ubuntu machine but an Arch Linux box running Gnome, but you should be able to find an older version of flash or flash-sse for Ubuntu.
What will happen is that the newer versions of Firefox (I have v27) will block older flash versions automatically with a security warning. You will have to click Activate ... and then Allow ... to view flash. Personally I prefer this as I don't like flash adverts streaming to my machine and using bandwidth.
The security risk, if any, can be mitigated somewhat by sending file locations to &>/dev/null, so that all data streaming in will be shredded on arrival.
[update] I have just installed Chromium v32+ and the above setup (Shockwave Flash 11.1 r102) works for Chromium too - without the grumbling that Firefox exhibits.
For Ubuntu you might be able to find a Debian package that will work. A good link to follow is https://wiki.debian.org/FlashPlayer. It seems that Flash 11 is the last version to work on Linux.
Otherwise Use HTML5
Alternatively, you can enable HTML5 instead of flash on any HTML5 capable browser (especially Chromium) by following this link https://www.youtube.com/html5 and clicking Request the HTML5 Player. From this point forward your Youtube videos and some others will play on any HTML5 capable browser, but I found on my older machine this was too resource intensive. On newer machines I don't think this will be a problem. This is where all streaming content is headed anyway and is why Adobe Flash will be phased out in the longer term.
If you search for flash player alternative only because the adobe flash plugin for linux is outdated, you could install the latest version of the adobe flash player through pipelight which allows you to use its supported plugins nearly as if you use windows browser.
then you could keep the plugins up-to-date using sudo pipelight-plugin --update
Best Answer
Indeed for the latest flash you will need chrome or chromium with pepperflashplugin-nonfree package (chrome and pepper flash are available for 64bit only).
However firefox uses flash player which is old and it may have bad performance but it still gets some security updates so it can be considered secure enough.
I always prefer chromium because I believe it's faster (however with greater ram usage) but both browsers have excellent security.
Just keep in mind that flash is going to die sooner or later so don't care about it. Most sites use html player instead of flash nowadays (remember that android doesn't support flash at all) and I read somewhere that google plans on blocking flash player on every site by default in a few months (user will have to explicit enable flash in a specific site).
So I believe you can still use firefox with no problems at all if you like it.