"Speaking" Korean without really speaking Korean, as a tourist in South Korea

Is it easy to learn?

No, it is not easy. Korean kids struggle all the way until they get into university. You stand no chance 😉

Learning the alphabet is, in fact, quite easy. Reading anything of value is quite so very hard. In fact if you wanted to read any respectable newspaper, you better learn Korean and Chinese. Majority of fancy words are borrowed, most either from Chinese or English, and words of Chinese origin don’t quite follow same rules as native Korean words when it comes to compound noun composition or even pronunciation and grammar. Most place names can be written in Hanja, which implies that aspect of pronunciation.

Will you need Korean?

Unfortunately, it’s quite a lot of hit-and-miss; For example:

• intercity buses are awesome in Korea; a clerk at the bus terminal in Seoul spoke enough English and understood me fine, and I suppose if they didn’t, they’d find a colleague who would; Anyhow I could get a ticket to some town of interest. But buying a return ticket in that town was quite a hurdle, there was only one clerk, who only spoke Korean. Neither did they accept credit cards.
• subway signage is bilingual, but just try to comprehend a bus route at a bus stop. Refer to http://www.ideacode.eu/seoul-glimpse-into-the-public-transportation photo with route 603.

What can you do in practice?

Read up on customs and learn a few basic phrases, so that even if you cannot communicate, you would be polite and score points for having tried. It is really appreciated.

I think jpatokal gave excellent advice – considering how much you asked! That’s a lot of motivation for a mere tourist! May I add (I live in Korea):

When you try to communicate in English, be patient: they may understand you, but they will need their time to respond to you if they are not very fluent.

Have a pen and paper ready, or type on your smartphone. People are usually better at understanding a foreign language in written rather than oral form. This is advice a Japanese guy gave me before I visited Japan. That said…

Koreans may instead tell you to speak to their smartphone, which will then translate spoken English into Korean. Again, be patient.

I can tell you that google translate is very, very bad with Korean sentences. (Individual words are ok). Babel XL is a bit better, but in general most output text will be nonsense.

Have fun!

That’s a lot of questions, son, but I’ll give you a general rundown based on my experience.

• English signage in the major cities is sufficient for getting around, eg. the Seoul Metro and Korea Rail have all major signs and announcements in English (and Japanese and Chinese!), so you won’t need hangul for a visit of a few days. Major tourist attractions will have signs, pamphlets etc in English. But the beaten path in Korea is pretty narrow, and once off it (eg. a bus terminal in a rural town) Roman letters will be few and far between.
• English proficiency varies from adequate to nonexistent. As in Japan, few Koreans get much practice in speaking the language, and will understand written English better. That said, I don’t remember ever having running into much trouble: if picture menus and plastic food models aren’t enough, sign language and pointing works wonders at restaurants etc.

• Outside major cities, or for a longer stay, knowing hangul will come in very handy. It’s a surprisingly logical and consistent system and thus not particularly hard to learn (especially when compared to eg. Chinese), but it is also radically different from any other writing system and takes a fair amount of practice because everything looks so similar at first glance. For example, here are the vowels of hangul:

  ㅣ ㅔ ㅚ ㅐ ㅏ ㅗ ㅜ ㅓ ㅡ ㅢ

  • …but even simple pattern recognition bits will help: if you known a circle at the bottom of a block is read -ng, you can already tell Pyongyang (평양) apart from Seoul (서울).
• Once you’ve got the grasp of basic hangul, you’ll also learn to pick out many English loanwords, since they tend to have lots of null consonants and schwa-like eu vowels that show up as dashes at the bottom: “stress” is 스트레스 (seu/teu/re/seu), “ice cream” is 아이스크림 (-a/-i/seu/keu/rim), etc. Compare with the dense blockiness of a native Korean (well, Sino-Korean) word like “airline” 항공 (hang/gong). But the difference is not as clear as in Japanese katakana vs kanji/hiragana.
• A few stock phrases will smooth your way (hello, thank you, yes, no, etc), and having some clue about how to pronounce place names is of course helpful: bone up on Revised Romanization if you get a chance, so you don’t trip up on all those eo and eu sounds. (Repeat after me: Gyeongbokgung, Changdeokgung, Changgyeonggung, Unhyeongung!) But a visitor cannot realistically expect to pick up conversational Korean in a week or two.
• Korean pronunciation is kind of tricky for the Western speaker, although (again) nowhere near as hard as Chinese. Wikivoyage has a good intro, but TL;DR, some of the vowels are unusual and most consonants come in three separate flavors (unaspirated, aspirated and tensed), the last of which does not exist in English.