So as the title states, I'm wondering how it works if I were to bring some snacks (mostly chips/crisps, drinks, candy) from the US all the way back to Sweden. This is my first trip flying, so I'm pretty curious how it would work. Would I need to declare my snacks and stuff at the location where I pick up my suitcase (which will be in Denmark), or do I need to notify them when I check the suitcase in the US before the flight?
With importing personal amounts of food, as you are doing, the main customs concern is not whether you need to pay duty on it, which is quite unlikely, but whether it's allowed into the country (or the EEA) at all. Depending on the specific food and its origin, any specific item might be banned for import or restricted. This usually isn't a problem for pre-packaged food from the US, but it is likely to be a concern if any of the items contain meat or dairy (e.g. beef jerky, sausages, fish, cakes, etc) or unprocessed plant matter such as fruits and vegetables. Most countries want to inspect such items and you should always declare if you have meat, dairy, fruits or vegetables.
If you aren't sure, you can always go to the red channel and tell a customs officer what you have and ask whether it is allowed. The general rule with customs is that if you declare something and it is not allowed, the worst that happens is you lose the item. But if you don't declare something that should have been declared, and you get caught going through the green channel with it, for instance via a random check, then you could receive a significant fine and possibly lose the item, even if the item ultimately would have been allowed to import if you had declared it.
In some airports there is not actually a red channel to walk through, as there is a green channel (and in the EU, a blue channel for intra-EU travel), but there will be a red phone to call customs and/or a customs officer waiting nearby.
Where you go through customs can vary from trip to trip. In a few cases, a country has set up pre-clearance in the country you depart from, as when flying from Dublin to Boston on Aer Lingus; for this trip one goes through US customs and immigration in Dublin. This is unusual and I don't think it happens with Denmark. If you fly into a region with unified customs procedure, you may go through customs when you arrive at the first airport in the region. For example, flying from Boston to Dublin and on to Edinburgh, I went through customs in Dublin.
No doubt you will be allowed to import for free some items if the value is under some threshold; if all you are going to leave in the country are the snacks, it shouldn't be a monetary problem. What could be a problem is the agriculture regulations. The country where you go through customs will want to make sure nothing you bring in has any insects or diseases that could damage crops and livestock.
The simplest thing is to only bring as much as you can eat on the plane, and throw anything you don't finish in the trash before you leave the plane. Bring a small amount of cash to buy food and beverage where you arrive, so you won't be hungry if you run into any problems getting your credit card to work. Ideally the cash would be the currency used in whatever country you plan to eat.