I'm running a homemade campaign for a 2nd level paladin, a 1st level sorcerer, and a 5th level fighter. I was rolling for random encounters, and satyr attacked. They dealt with it in 2 rounds. I credited it to good dice rolls. Then a bugbear attacked, and before it could even strike, the paladin dealt exactly enough damage to kill it, in one roll. Seeing how the rest of the campaign consists mainly of goblinoids, I don't know how this will challenge them. Any thoughts?
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You should read the section in the DMG about creating encounters. There is a whole section on evaluating encounter difficulty, and another about how to increase difficulty.
The free basic rules have the section on combat encounter difficulty, which has some basic guidance on how to put together an encounter that is easy, medium, hard, or deadly for your group. You've got a big level imbalance, which is going to make this particularly difficult — that 1st-level sorcerer is going to be incredibly fragile. The book cautions:
For example, an ogre has a challenge rating of 2, but it can kill a 1st-level wizard with a single blow.
I'd suggest handing out 900XP to your characters pretty quickly, to get past this problem, maybe just for free, even. It'll be a lot easier to deal with, say, two 3rd level characters and the 5th level one. But even at the levels you've got, the encounters you've described are very easy — the guidance suggests 350 XP worth of monsters is an easy encounter, and a hard encounter would be 975 XP.
A satyr is a ½ CR creature worth 100 XP. So that's definitely down below easy. If you want to make this interesting, you need at least two satyrs — that's 100XP each and a 1.5× adjustment for having two creatures, so 300 XP... still easy. Five would bring that up to 1000 XP (with the higher adjustment for an even larger number of creatures), so I'd expect that to be more interesting — although, again, be sure to be careful of the unbalanced party hardiness.
A bugbear is a little tougher — CR 1 and worth 200XP — but you can still see that alone is still below "easy". It's not at all surprising that the paladin could drop one in a single round, if they get very lucky and use limited-per-day options like Divine Smite.
It sounds like you're basically just throwing single monsters at your party. Don't do that — at least not every time. Goblins, alone, are very weak CR ¼ monsters worth only 50 XP. You need 7-8 of them to make a hard encounter for your party as it stands now.
It's also interesting to have mixed groups of enemies — a goblin war party with a hobgoblin captain is an easy one. Throw in some spellcasters — maybe a hobgoblin devastator from Volo's Guide, or make your own with class levels. Goblin war cleric... or, you know, everyone loves a bugbear bard, right?
You can also make the attack happen in an interesting situation. The DMG suggests, for example:
The characters are hanging from a rope, in the midst of scaling a sheer wall or cliff, stuck to the floor, or otherwise in a situation that greatly hinders their mobility or makes them sitting ducks.
Since you're focusing on goblinoids, pay special attention to the abilities that make these creatures unique. Just the plain goblin has +6 to stealth, which is amazing for a CR ¼ creature, and the Nimble Escape feature lets them move around in combat. And they've got darkvision, which your party may not. Have goblins ambush in a situation where they can take advantage of all of this. Bugbears have Surprise Attack — again, playing towards ambush. On the other hand, Hobgoblins have Martial Advantage, making them quite deadly when working with a group.
Remember that the encounter difficultly system is guidelines, though, not a complete algorithm, so take these situational things into account when judging difficulty.
Difficulty doesn't just need to mean combat strength, too. For example, for your satyr encounter, maybe throw in a band of pixies or sprites angry that you're trampling through their woods. Unlike the chaotic neutral satyr, they're good-aligned, but they can still be very angry and offended. Maybe it starts as a combat encounter but can be turned around — or starts as not combat, but someone gets twitchy with a tiny shortbow, and... then what happens?
On another note, take into account that the game assumes more than one encounter per adventuring day. This is from the same chapter of the DMG, or the basic rules:
Assuming typical adventuring conditions and average luck, most adventuring parties can handle about six to eight medium or hard encounters in a day. If the adventure has more easy encounters, the adventurers can get through more. If it has more deadly encounters, they can handle fewer.
In general, over the course of a full adventuring day, the party will likely need to take two short rests, about one-third and two-thirds of the way through the day.
If your party is resting more than that, your individual encounters should be more difficult.
Really, the bottom line the same as my first one here: read that chapter in the DMG. There's stuff in there that's not in the basic rules about getting the balance right, and about how to make encounters feel interesting. Spending some quality reading time here will really help you get this right and help your group have more fun.