cryomancer build help.

[Amdy_vill]

I want to build up a cryomancer as a backup character. right now I'm playing in a low-level game. hopefully ill survive to 5th level but I think having a character built and ready is used. I want to do a dragon born cryomancer who focuses on battlefield control more than direct damage. I feel draconic soul sorcerer, metallic dragon born from fizzbins is a clear choice. with maybe a dip somewhere if it would help. I'm mostly looking for spells list suggestions and possibly dips/ alternative class picks for this idea. at the moment we are level 3.

[Greywander]

They've been adding a few more cold spells lately. Check out Rime's Binding Ice; it's a no-concentration spell that deals damage (save for half) and covers the targets in ice, dropping their speed to 0 until they use an action to remove the ice. Not bad value for a 2nd level slot. Makes for a nice standard blast-and-control combo, especially once you have more spell slots and can afford to burn a few 2nd level slots in one encounter. Other spells to consider are Ice Knife (1st level), Sleet Storm (3rd level), Ice Storm (4th level), and, of course, Cone of Cold (5th level). Also consider taking other spells and refluffing them to be ice-themed, such as Shatter or Spike Growth. Fog Cloud would probably be on-brand as well. Wall of Stone could become Wall of Ice, and Flesh to Stone could become Flesh to Ice. You've got some options. Not a lot, but you've got some.

[icedraikon]

Honestly, Transmuted spell would work wonders. If you've already picked your metamagics, you can take Metamagic Adept at 4th and grab Transmuted + Subtle or w.e else you want. Wall of Fire transmute to cold, so Wall of Cold, could just be a bunch of ice spikes whirling around or something. Fireball? Discount Otiluke's. Best part? Only 1 sorcery point.

If you can talk your DM into adding a homebrew spell list similar to those added in Tasha's, that would be ideal.

[LudicSavant]

The most I ever found myself using ice spells just because they were natural good choices rather than to force a theme was as an Evoker.

Why? Because they have some really fantastic synergy with many of the game's best ice spells.

Cone of Cold's gigantic conal area attached to your character is often a problem for allies in the way... not so for an Evoker who simply Sculpts the spell. Not only that, but it happens to be a great choice for Overchannel.

Rime's Binding Ice is one of the best 2nd level spells, but even more amazing with Sculpt Spells and Empowered Evocation... especially because it too has a big conal area (even if not nearly as big as Cone of Cold). And at high levels, you can use it at-will for Concentration-free damage and control!

Otiluke's Freezing Sphere can not only be used to nuke a truly massive-sized Sculpted AoE to clear out thoroughly-spread out foes, it also can be thrown by minions like familiars or animated dead, which means you can set up a helluva alpha strike with it. It also has a considerable range, able to hit foes 360 feet away. Its massive area means that Sculpt Spells is a significant boost to its usability, and Empowered Evocation applies as well.

Wall of Ice is a lovely hybrid spell that also has interesting interactions with Sculpt Spell. It's sort of a combination upcast Lightning Bolt, followed by an opaque wall (which means they can't, say, Misty Step out like they could with a Wall of Force), followed by a hazard spell. Importantly, with Sculpt Spells, allies can walk through the hazard unharmed. So you basically hit them for the shapeable 10d6+5 AoE, have them waste time breaking through the wall, take another 5d6 walking through, and then... can just have your allies switch sides with them, putting the wall between you and them again.

Dragon's Breath is a good way to give minions a breath weapon for Wizards, and cold is one of the several elements it can use. Have your owl do flyby strafing attacks, swooping in to deliver a cone then flying back to cover somewhere!

Evokers in Tier 3+ can make better use of the Frostbite cantrip than anyone else, because it applies both Potent Cantrip and Empowered Evocation. 3d6+5/save for half with Disadvantage on an Attack ain't too shabby for a cantrip. And of course Ray of Frost is available too, and applies Empowered Evocation. Toll the Dead also synergizes particularly well with Potent Cantrip, and while it might not technically be a cold spell, I feel it's still vaguely on-brand.

Fire Shield also has the option for a cold shield, and is a good non-concentration spell with sufficient duration to last multiple encounters. It can also be used with Contingency and Overchannel, using resources from a different adventuring day. If you go with a Hexvoker, it also can apply Hexblade's Curse on every hit.

Armor of Agathys can be gotten by either a Hexblade dip (which is especially good for Evokers, enabling builds like the Nuclear Wizard or Retvoker), or from one of the better Wizard races (Eberron's Warding Dwarf). It too synergizes well with Contingency. And this + Fire Shield + armor from a Hexblade dip means you can make a cryomancer tank that's suicide to attack in melee, if you're so inclined.

Illusory Dragon is a solid 8th level option that deals Cold damage among other things... and isn't on the Sorcerer list for whatever reason.

There's also a lot of spells that, while not technically dealing Cold damage, still kinda fit with that northerly feel (like spells of wind and darkness and fog and sleet and slow and such. Even something like Wall of Force could be fluffed as a completely transparent pane of molecularly perfect, unbreakable ice).

[Sorinth]

I would add that you should also ask your DM about refluffing spells without changing the mechanics. For example Hold Person could be flavoured to be encasing the target in ice with no mechanical change to the spell. Or Maximillian's Earthen Grasp if it's made out of snow/ice instead of compacted soil but mechanically is the same and so still does Bludgeoning damage.

[Evaar]

The most I ever found myself using ice spells just because they were natural good choices rather than to force a theme was as an Evoker.

Why? Because they have some really fantastic synergy with many of the game's best ice spells.

Otiluke's Freezing Sphere can not only be used to nuke a truly massive-sized Sculpted AoE to clear out thoroughly-spread out foes, it also can be thrown by minions like familiars or animated dead, which means you can set up a helluva alpha strike with it. It also has a considerable range, able to hit foes 360 feet away. Its massive area means that Sculpt Spells is a significant boost to its usability, and Empowered Evocation applies as well.

Just adding a note that Evoker might be the only class who can really use the delayed/thrown version of Freezing Sphere well. The effect has a 60 foot radius. You can only throw the stone 40 feet. So by necessity, the thrower will be caught in the effect. The Evoker cannot exclude herself from the effect, but can exclude anyone else. Just more reason it's uniquely better for the Evoker than other casters.

[Melphizard]

As usual with most [X element] build ideas, Scribe Wizard is gonna be your easiest route to getting cold damage whenever and wherever you need it. With Scribe you can do Icy Fireballs and Chilling Steel Wind Strikes.

Alternatively, as Icedraikon has already mentioned,
Transmuted spell would work wonders.

Past this there isn't a lot more I can offer aside from maybe checking out the subclasses in Knuckleheads & Other Such Curiosities, which was an Adventurer's League addon for Rime of the Frostmaiden. Ofc Adventurer's League didn't actually let us use any of the subclasses (I crave the Brutality Monk) but they're still Icewind-Dale-themed so something in there may prove to work. Heck, one of the subclasses in there is a Sorcerous origin called Frostblooded.