No need for an introduction her. Let's get straight into some nice comparisons.
The verb at sørge translates as to mourn or to feel deep pain for something. So when you sørger over something, this means you are feeling pain because of something that has happened or something someone did.
Han sørger over tabet af sin kæreste (he is mourning the loss of his girlfriend)
So what about sørge for? This can be thought of as “to take the pain for” something. Or in a more logical way, to take care of something, so someone else doesn’t have to suffer it.
Han sørger for maden (He is taking care of the food)
Jeg skal nok sørge for, at han ikke drikker for meget (I will make sure he doesn’t drink to much)
Nothing much to say here. Just remember, if you're talking about something in the future, use glæde til:
Jeg glæder mig til at rejse (I´m looking forward to travelling)
..And if you're talking about something which is currently making you happy, use glæde over:
Børnene glæder sig over, at deres farmor også er her (The children are happy that there Grandmother is also here)
When around the dining table, you can use bede om to politely request your fellow diners to pass you something:
Må jeg bede om vand (Can I get water, please?/would you pass me the water, please?)
If you are in a bakery or store, you can also use bede om for any request or wish. Just as you would Jeg vil gerne have... (I want/would like)
Jeg vil gerne bede om et stykke chokoladekage (I’d like a piece of chocolate cake, please)
The phrases is commonly used when refering back to something you asked someone to do, or something someone asked you to do.
Han bad mig om at tage opvasken (He asked me to wash the dishes)
At bede til means you are asking for something internally, usually in a concentrated way to God (to pray).
Han beder til Gud (He his praying to God)
And as a side note, at bede på, would be the way in which you ask or pray: Hvad er den rigtige måde at bede på? (What is the right way to pray?)
The verb at slappe means to loosen and is used for physical things such as a rope or a belt.
Han slap tovet (He loosened the rope)
The phrasal verb at slappe af translates as “to relax” = to physically relax your body due to exhaustion or just to relax and enjoy yourself.
Jeg har trænet i en time og er altså nødt til at slappe af (I have trained for an hour and really need to relax)
Vi slappede af og hyggede os i sofaen (We relaxed and enjoyed ourselves on the sofa)
It can also be used to express mental stress, tense or worry.
De bekymrede forældre slappede helt af, da deres søn endelig kom hjem lørdag eftermiddag (The worried parents completely relaxed when their son finally arrived home Saturday afternoon)
In this way, it relates to the phrases tag det roligt and bare rolig (take it easy/don't worry) when the imperative form is used: Slap af! (Relax!) Which can be physically (you need to relax your body) or mentally (Relax your mind/don´t worry).