The cantrips listed as known on the character sheets are correct.
Page 23 of the Starter Set rulebook is a list of cantrips that exist in the Starter Set, not a list of cantrips the cleric and wizard know. A 1st-level wizard or cleric knows only three cantrips (though the wizard in the Starter Set has an extra known, as that is a bonus high elves get). More are listed because you do learn new cantrips by levelling. Both the cleric and the wizard learn one new cantrip of choice at 4th level. This is detailed on the backs of the sheets rather than in the levelling guide.
Additionally, having a reference for cantrips not yet known by PCs is useful to the DM if they appear in play when cast by characters other than the PCs, or if found on scrolls in treasure, etc.
You're right. It's only base dice, no modifiers.
This probably sounds terrible, however, there is a good reason:
It's more balanced than it seems.
It's hard to tell sometimes; believe me I know. But, until you get more hands-on experience with the game, you have to give the game you're playing a "grace period" where you trust that the designers made the correct decisions, even if it seems like they didn't at first. Every campaign you play or DM is going to be different. If you change players or DM, then the campaign will be completely different. Some of those games, you will have players that built weak casters, and experienced grognards that built martial types. Although it is a roleplaying game, the "player skill" of the game comes in with decision-making during game time, and character building during creation.
That said, if you GM a game and you consistently encounter the same balance issues across encounters, then yeah, you should feel confident that houseruling whatever the issue is will improve your experience.
Your Wizard's spell slots are pretty big.
Bear with me a moment and imagine that the Wizard class is actually Batman. The biggest strength of the Wizard is that they have a tool ready for every situation -- at least, if they're a good Wizard. That's the player-skill-decision-making part. The spells your Wizard picked aren't bad, they just make him very good at things that aren't combat. Your Wizard has a ton of tools for dealing with non-combat situations. Imagine if Batman didn't bring his Batarangs when he went out on a mission. He's still the best detective on the planet, but now he isn't as effective as he would be in a fight. Spells are really, really good at solving specific problems.
That said, given the nature of the encounters you described, you might try throwing your Wizard a bone if you haven't already. D&D is more than just combat, and if you aren't giving your non-combat specialist Wizard any non-combat to specialize in, he'll feel like a useless player. If your player is concerned about his build, then you can allow him to switch out a cantrip, or give him a couple of good AOE scrolls in the next loot pile to scribe.
In addition, cantrips scale at certain levels. It's not immediately, but you can be sure that the cantrip will be a good standby at later levels.
Best Answer
It is indeed the case that Ray of Frost does 1d8 damage, while a comparable weapon might do 1d8+ability modifier damage. However, since you can use your spellcasting stat (assumed to be your highest ability), you will have a better chance of succeeding in your attack roll. In addition, Ray of Frost slows on a hit, which is a powerful additional effect.
I didn't find any rules text that states "Don't use the spell description but some other rule similar to weapon attacks", but for reference, here's an excerpt of the section about weapon damage rolls in the How to Play:
And Ray of Frost, from the final playtest:
As nitsua60 noted in a comment, there are certain class features that allow a caster to add her or his spellcasting modifier to a cantrip, e.g. the Warlock's Agonizing Blast. If any caster could just add the modifier, this feature would be pointless, or would at least be worded differently (add double your modifier or so...).
So yes, the ray does a bit less damage, but with the control effect it is equally good if not stronger than a 1d8+3 weapon attack - and you can use your primary stat for attack rolls, which could easily be a +1 or +2 on the to-hit.