It's really a loaded question, as Kevin2 implied there's the current W3C recommendation of HTML 4.01 Strict... but there's also the SECOND recommendation of XHTML 1.0 STRICT.
I prefer XHTML 1.0 STRICT due to it's more consistent structural rules -- but if you can't "stay between the lines" by closing 'empty' tags (and understanding what the specification actually means by an EMPTY tag -- <div></div> is NOT a empty tag by the spec, even if it is empty) you may be safer with HTML 4.01
... and of course regardless of whether you choose HTML or XHTML you want STRICT so the validator will chew you out when you use elements and attributes like CENTER, FONT, BGCOLOR, BORDER, TARGET, EMBED, etc, etc that have ZERO business in any website written after 1997.
HTML 5 while 'trendy' should be avoided unless you REALLY need to use one of it's legitimate improvements (MANIFEST and... uhm, well.... there's... uhm... wait what about... ok, there's MANIFEST) or have to use the idiotically redundant AUDIO or VIDEO tags because crApple users don't actually own the hardware they bought and freetards raging against the machine have in the name of fighting vendor lock-in managed to promote vendor lock-in.
Even if using those, I would still validate as 4/X1 STRICT first! Live with the errors the two or three HTML 5 tags you have to use, then at the last moment before deployment slap the 5 doctype on it and validate again; usually if you can make it validate in 4/X1 Strict (excepting of course the two or three HTML 5 elements) it should validate in 5 because:
1) HTML 5 is a superset of HTML 4
2) HTML 5 validation is a joke thanks to the loosening of the structural rules to the point that document structure becomes meaningless. Stands to reason given that HTML 5 seems designed to encourage coding that reeks of the worst of pre-STRICT practices and dials the clock back to the worst of 1997-style code.
Basically, if you HAVE to use 5-crap, design in 4 STRICT first, and do yourself a huge favor and forget the pointless code-bloating redundancies like SECTION, NAV, HEADER, FOOTER and ASIDE even exist.
Though looking at your site, it's a bit late for that... You've got HTML 5 code-bloat in there with no logical document structure -- so you should have the 5 doctype on it; that or toss it and start over, which I'd suggest from the broken layout anyways.