így and úgy work similar to ilyen/olyan, in that they are used to describe or emphasize something. However, the difference is that ilyen/olyan are used with nouns.

így / úgy are adverbs that pair with hogy(an) (how/in what manner?) and are used with verbs.

Depending on the sentence structure, you will place így / úgy either before the verb, or after the verb, but they will often stay together. 

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Alex’s Notes 

There are some things that can go between them, like azért in the sense of ‘after all/though’ or nem/sem/mégsem. There might be others but these were the ones that occurred to me. 

Examples 

Így mégsem mehetek bulizni, előbb hadd csináljam már meg a hajam (I can’t go to the club like this after all, let me do my hair first at least). 

Azért is a bit less frequent. It’s definitely nicer/more proper to say: 

Azért nem kéne úgy hagyni a szobádat, mint ahol tornádó pusztított (You shouldn’t leave your room looking like it was devastated by a tornado, though) 

But you’ll run into “Úgy azért nem kéne hagyni a szobádat, mint…” in general usage. 

Nem/sem – words of negation, in general, MUST go in between the adverb and the verb

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így = like THIS, in THIS manner/way

így follows the same rule as ilyen: it is generally used to describe or emphasize something (in this case, an action) that is observable and close to you. In temporal (time) terms, it is usually an action happening right now.

Examples

(a parent scolding their child for misbehaving in public)
Miért viselkedsz így? Hagyd abba, vagy/különben¹ hazamegyünk
Why are you behaving this way/like this? Stop it, or we’ll go home.

¹Vagy and különben mean the same thing, you could use either, but in this particular case most parents will use both in this specific way 

(teaching someone how to cook something)
Akkor most félbe vágjuk, így…
So now we’ll cut it in half, like this/in this way…

(your friend is over or under-dressed for the occasion and you say to them)
Így jöttél? Oké, ahogy akarod…
You came like this? Okay, however you like it…

(during an argument with someone)
Nem akarok így vitatkozni…
I don’t want to fight like this…

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úgy = like THAT, in THAT manner/way

úgy follows the same rule as olyan: it is generally used to describe or emphasize something (in this case, an action) that is NOT as close to you, but can still be observable. It is usually an action that is happening either in a different location, or has happened at a different time. In some cases, it can still be an action you’re currently observing (it depends a lot on context and situation).

I am going to slightly change the sentences from above to show the circumstances in which it could change to úgy.

Examples

(a friend is describing how her child misbehaved in public, and you think)
Ha én úgy viselkedtem, onnantól anyám ki sem engedett volna a házból
If I behaved like that, my mother would never have let me out of the house again

(you’re watching a cooking show with a friend, and point out one of the actions)
Nézd, úgy¹ csinálod²…
Look, you do it like that/that way

¹így could also technically still work here, if you’re currently watching the action

²You would use definite conjugation here (as there is technically an invisible ‘azt’), and úgy csinálsz/úgy tesz usually means “to pretend”

(you’re looking at a picture of a friend who was under or over-dressed for a previous occasion, and you think)
Úgy ment oda? Oké, ahogy, akarja…
Xe went there like that? Okay, however xe likes it…

(your significant other is telling you about a fight he saw a couple have in public as he was walking home, and you say)
Azt remélem, hogy mi soha nem vitatkozzunk úgy…
I hope that we never argue like that…

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“így/úgy“ with mint

You can use így / úgy with mint (like/as) to make a comparison as well. Typically, úgy will be used, but in cases when the thing that is closer to you (either in time or space), or the thing you’re pointing out/focusing attention on is what you are comparing TO, then így can be better.

(you’re sitting in a park and watching a runner who has passed by several times already. you remark to your friend)
Bár én tudnék így / úgy futni, mint ez a lány (a comparison)
I wish I could run like this girl
literally: if only I could I run like this/like that/in the same manner, like this girl

Soha nem tudtam így / úgy¹ énekelni, mint az anyám (I‘ve never been able to sing like my mother)
literally: I’ve never been able to sing like that/in that way, like my mother)

In this case, it depends on the situation whether így or úgy would fit better. If your mother is in the same room or you otherwise just witnessed her singing, or if someone had just talked about/described your mother’s singing, then így would work. If your mother is in a different location, or no longer alive, or you are bringing the topic up without it being mentioned previously, then úgy would work better.

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“úgy“ with csinál/tesz

In some specific occasions, you can say “úgy¹ csinál/tesz…” to show someone how to do something, but more commonly this pairing is used to mean to pretend to be/to act like. You will generally see it followed by “mintha” (like/as if).

¹In this particular sense/usage, it’s always úgy and always using the indefinite form of the verb!

Úgy csinálsz, mintha barátok lennénk
You’re acting like/as if we are friends
literally: you are acting/doing like this, as if we would be friends

A gyerekem néha úgy csinál, mintha a legrosszabb anya lennék a világon
My kid sometimes acts like I’m the worst mother in the world
literally: My kid sometimes acts like/such that, as if I would be the worst mother in the world

Ne tégy úgy, mintha soha nem segítenék neked
Don’t act like I never help you
literally: Don’t act such/like that, as if I would never help you