There are some 'issues' in the Southern Thai states I believe (think it's the ones that border Malaysia) so you might want to check state department advisories on this. You could however easily get the train from Singapore to Butterworth, ferry over to Penang then hop an LCC up to HKT. From there you can get a bus up to BKK, another LCC or bus part-way then pick up the train again.

In '93 I did an overland trip SIN-BKK and back again. Stopped at a place with hot springs at the south end of the Thai/Burmese border - Ranong I think. Absolutely fascinating as it was very much off the beaten track - imagine it's very different today - looking at Google Maps it looks like a massive place, not the dusty village I recall...