That is: turn each lash adjuster counterclockwise half a turn and see if anything changes. If so, do a proper valve adjustment, pay attention to the tune-up parts and technique suggestions in this post, and be happily on your way.
It may turn out you do need major engine work, in which case you're on the right track as far as the cylinder head goes (hardened exhaust seats, good guides, etc). Putting a nice, good-sealing head atop an old bottom end is a sure-fire recipe for an oil burner de luxe, so you're also on the right track pulling and going through the whole engine. This will involve the usual thorough cleaning (inside and out—there's probably quite a bit of rust and mud in the coolant passages) and extensive careful measurement to determine what-all is needed to bring it up to snuff, so ignore bad advice to just blindly reuse the original bearings and otherwise like that. Likewise, failure to hot-tank the head and the upper valvetrain components would be foolhardy.
There is also no good reason to go chasing all over for a NOS shim steel head gasket—the current production composite types make it easier to get a good block-to-head seal for everything (combustion, oil, coolant), and if you wish to compensate for the minor increase in gasket thickness you can have the head and/or block surface skimmed.
New timing chain and sprockets, yes, and you might want to take the opportunity to put in a № 2106R camshaft from Oregon Cam—that's the Dutra RV-10RDP, which breathes much better than the '63 stock cam for peppier performance and better economy while keeping smooth idle and driveability.
Will the oil pump want refurbishing? Maybe or not; this is another area to measure, not blindly assume. Either way, though, it will be a good idea to service the oil pressure relief valve to make sure it's doing its job, as described in these two threads: thread 1, thread 2.
It's your choice whether and how to paint the engine once it's all put back together. International-Harvester Red is an excellent match for the original color on a 1963 225; the Mopar red engine paint is not.
There's any number of other ancillaries to pay attention to—if you still have the original bolt-together fuel pump, it's a fine idea to put a rebuild kit in it; those come from Then & Now Automotive. If you want or need a water pump, shop carefully because there's a lot of junk on the market, etc.
One way or another you ought to be able to win the fight with that Stromberg carburetor unless it has been horribly abused; it's a very well-designed unit. Because it's a one-year-only item, rebuild kits are difficult to find, but not impossible; it's a kit № 233 from thecarburetorshop.com.
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