I have been there twice from Australia.
Buy the rail pass online before travelling. It will save you heaps and importantly give flexibility. See the Korail site.
There was an excellent themed tourist train my wife and I travelled on from Busan (on a Sunday?) towards the west of the nation.
The high speed trains are extremely frequent but a lot of the route is (as one would expect) in tunnels or has noise barriers.
Don't just visit Seoul but Busan as well. There are excellent hotels adjacent to or near Busan that shouldn't break the bank.
Mokpo and Yeosu are also worth a visit. In one of them, I walked down a road lined with cheery blossoms: I think Yeosu, but unfortunately need to verify.
I also went on the train to the last station before the DMZ, but did not organise a day tour (latter are all by bus from Seoul if I recall).
Some hotels will not admit Westerners as they're for a distinct purpose., although not all in the category do this. Suspect I don't need to elaborate.
Remember there's the all stations airport train that is far cheaper than the KTX Express or whatever it's named. I bought my Seoul Metro ticket IIRC at Incheon Airport on arrival from a convenience store near the platforms.
You'd have to be unlucky to have any problems with theft or other nuisances. It's an orderly country (although those who sadly were caught up in the crush in recent months would beg to disagree).
I found the availability of fresh fruit to be scarce, but otherwise, no problems with food.
Quite a lot of USA military personnel were around Seoul station.
The Seoul and Busan Metros are worth travelling on, with the latter serving a beach. Have a look at TA (TripAdvisor) for things to do as we travelled on a gondola IIRC in Busan, easily reached by metro and bus.
If I find my notebooks I'll be more definitive.
Like Taiwan, it's an underrated country for rail enthusiasts.
Combining SK with Philippines can give rail in one, great white sand beaches in the other, but your travel plans are obviously fixed.