Is this pretty typical, or have I missed something?

Yes, just as in many good games (tabletop or otherwise), you are given a finite resource which you must learn to economize. It's all part of the game!

At low levels, it's pretty easy to run out of spell slots, you only have 2 of them so it is up to you to manage them and make the most out of what you have. Even as other spellcasting classes such as Bard and especially Warlock, you really need to think about what sort of situations you want to spend your precious 2 1st-level spell slots on.

Without spell slots, you have other options for offense but going into melee with your armorless self is less than advisable so your best bet is to use your cantrips just as you are doing right now. They might look a bit underpowered to you at first, but that is because these level 0 spells give up damage for a slight mechanical benefit. Shocking Grasp prevents your target from taking reactions(thereby allowing you to get away safely), Ray of Frost slows down your target by 10 ft, Fire Bolt sets things ablaze, even Sacred Flame negates cover. These cantrips also increase in damage as you reach higher levels (usually at 5th-level so it is still a ways off)

Don't worry, 2nd-level will come faster than you might expect. Then you'll have 3 1st-level spell slots to contemplate spending.

As a final note, and this may be nothing but if you find yourselves struggling for resources every session, remember: a long rest isn't a "session ender", you can take long rests in the middle. In my group, we have fun over the logistics like bedrolls and sharing them because one guy didn't have one.

There are also guidelines in the DMG on the experience earned in an adventuring day, found in page 84. I felt like I should mention this because it's odd that you've had 5/6 encounters and still haven't gotten to 2nd-level.

There's nothing RAW that indicates that players would know whether or not a creature is affected by most spells, though presumably in the instance of Ray of Sickness, players could infer whether the spell had taken effect based on a creature's behavior.

Does the hobgoblin captain slump for a moment, resting on her spear? Does the shadow demon react at all to the beam? Those are probably telling.

Zone of Truth appears to be a little different since the caster explicitly knows whether the target succeeded or failed the saving throw, which can only occur if the creature is affected by the spell in the first place; a creature which isn't affected by the spell never makes a saving throw. In this way, the caster would know if the hostage weren't affected, if only circumstantially; instead of having succeeded or failed to resist the spell, the Rakshasa would effectively be invisible to the spell, which would surely be suspicious.