As mentioned above, Korea would best be discovered in the company of at least an ethnic Korean companion. Seoul, especially, can be dreary (and frighteningly expensive) for the solo traveller, but quite fun when properly hosted.
July is monsoon season in Korea, humid and hotter than Japan's June monsoon, but not as systematically overcast. The western coast is pleasant at that time; renting a condo by the beach, which could serve as a base for day trips, is indeed a good idea.
Cheju-do, the Korean Hawaii due to its scenery and weather, is the favored honeymoon destination. Expect higher prices and more developed tourist infrastructure than on the mainland. I am not sure how much down-to-earth local culture subsists these days.
Budget-wise, there is a clear differentiation between the international class accomodations catering to most FT posters, and the omnipresent inexpensive inns and guest-houses used by the masses.
The
KNTO website is a good starting point for general information and links.
<font face="Verdana, Arial, Helvetica, sans-serif" size="2">She was thinking of renting a motorcycle...</font>

In all of Asia I cannot think of a country where two-wheeled vehicles are deemed less suitable for women. Most Korean women do not know how to ride bicycles for a reason: culturally because the posture is considered undignified, more recently because bicycles and mopeds are viewed as 3rd worldish means of transportation, a no-no in image-conscious modern Korea. A foreign woman riding a motorcycle in Korea would certainly be exposing herself to potential abuse.
Other options: the extensive long-distance bus system is inexpensive, despite a middling safety record. The Korean rail pass is a steal; most Korean trains use older US-style rolling stock, with the TGV slated to start limited operations this year. Domestic flights are inexpensive.
A car would indeed be the best way to explore the countryside, but require an understanding of the Korean language and peculiar road laws (ever tried to take a left turn in Seoul? keep turning right...)
Road signs are mostly bilingual throughout the country, but spellings are not standardized, which can be confusing. Fine can be paid directly to the policeman
The 2-lane country roads meandering through the pine trees, red hills and lush rice fields in the summer are very soothing, though.
Within Seoul, forget driving. Even finding parking is a major problem. The subway is clean, inexpensive, and efficient. Busses are many, cheap, and difficult to master. The 70,000 taxis are inexpensive, if not as pristine as Japanese ones. All of this applies to Pusan, to a lesser extent.
Korea is not a particularly tourist-friendly country once outside the beaten path. In this oft-invaded country, do not expect the wide-eyed wonder a foreigner might expect in a south-east Asian village. Age, education, sedate behavior command respect, but brash/non-conventional demeanor (US military personnel is not particularly popular...) may arouse the ire of drunken local men if in the wrong place. Violent theft is not a concern.