This may be out of date since Myanmar is changing so rapidly these days, but as of July 2014 there are only two border crossings between southern(-ish) Thailand and Myanmar open to foreigners: Ranong/Kawthaung at Myanmar's southernmost tip, and Three Pagodas Pass north of Kanchanaburi, not really 'south' at all (it's actually a fair bit north of Bangkok) but still further south than the other two.
Now, Ranong/Kawthaung would be the obvious option since it's much closer, but last I checked (and Wikivoyage agrees) the road north to Myeik (Mergui) and Dawei is still off-limits to foreigners. Apparently it's theoretically possible to fly up, but this looks pretty flaky even by Burmese standards, and that's saying something.
So this means going via the Three Pagodas Pass is your only practical land option, despite the massive detour involved. Getting there from Bangkok is fairly straightforward, and there's supposed to be a bus onward to Mawlamyine, from where you can make your way south. However, even this border crossing is quite new and there are some reports of visa shenanigans, so due diligence will be required.
Of course, the sensible, comparatively expensive and arguably boring option would be to ditch the land crossings entirely and fly in via Bangkok and Yangon.