I'm not that great at this game, and only finished the beta once with an Adventurer mode Paladin. But I can do the first boss fairly reliably with half of the job classes, I think.
kohlrak: The first boss is making me his whipping girl (that's weird as a guy to say), and i'm reading that cross-class play is the way to go, not grinding.
Grinding is useful up to a certain point. Some classes I'm just bad at, and it gets harder after the 8th floor or so, but the first boss isn't too much trouble when I'm on a class I'm better at.
You might want to be level 6 for the first boss. Level 5 with only a few XP to go for the level-up heal would be even better, but 6 is fine, and 5 is probably doable.
When you get to the first boss, your first class might be job-maxed already, or close to it. If you switch jobs, you lose your higher-tier perks and don't get anything back for it right away, so wait; get weapon skills if you have extra JP. Immediately after Boss 1, you can grind the next two floors to get your second job running; 1000jp needed to activate 2nd tier perk.
Boss fight 1: For the first boss, you can throw traps and other consumable items around. Caltrops and kiting can keep stuff away, and other long-range damage stuff. It really depends on your class, and I'm bad at some of them, but for example Phoenix Wing gives you a few turns to do free attacks while running in circles. Anything that does damage repeatedly to multiple enemies will blow up the underlings. A flask and a meat kebab will give you a lot of health to work with, but you want to be at maybe 50% when you take a whole turn to eat. (It doesn't have to be a meat kebab, but that's one example of a higher-tier healing food).
Grinding: Up until this point, you should be able to conserve a lot of flask charges, and cap your bars with bananas and apples between fights if it seems necessary. Remember that your first heal per XP level at the town priest is fairly cheap, and the food vendor will restock as well; keep harvesting your trees too.
The lead-up to the second boss fight consists of many more powerful monsters, and I often find it hard to earn flasks and food faster than I use them around this part of the game. Grinding might be an important part of the equation. You can't grind Worthless monsters for XP or JP, but even Average groups can have dangerous Champions show up here and there.