There are a few classes you can multiclass into to gain skills, such as Bard, Ranger or Rogue.

In addition to that, in the player's handbook there is a feat called "Skilled" which allows you to gain proficiency in any combination of 3 skills or tools.

In Xanathar's Guide to Everything, we have the Prodigy feat which gives a skill proficiency and a few other things, but can only be taken by a Human, Half-Orc, or Half-Elf.

In Tasha's Cauldron of Everything, the feat Skill Expert gives proficiency in a skill and a few other benefits, with no racial prerequisite.

There is also a bard ability in the college of lore that allows you to learn 3 more skills at level 3.

Currently, the downtime rules allow you to gain proficiency in tools but not skills.

In case it isn't clear, you don't have to pick skills from your class skill list when you gain new skills through the "Skilled" feat. You can pick any skill you want.

The Short Answer (TL;DR)

You need Thieves' Tools to pick a lock but you don't need to be proficient with them.

The Long(er) Answer

You (usually) need Thieves' Tools to pick a lock or disarm a trap BUT you (probably) don't need to be proficient with them (maybe).

Exactly how you are meant to use Thieves' Tools isn't 100% explicitly and unequivocally covered by RAW (yet still) so this is my interpretation of those rules based on the rules I've found.

Part 1: Do you need Thieves' Tools to pick a lock?

This part of the question is easy: YES

Thieves' Tools are (surprisingly) a tool:

Tools

A tool helps you do something you couldn't otherwise do, such as craft or repair an item, forge a document, or pick a lock.

-PHB p.154/PBR p.50

This is pretty explicit and there aren't any contradictions to this general rule elsewhere. One potential problem is how to deal with improvised Thieves' Tools. The DMG has given us some good basic rules to deal with improvised Thieves' Tools under the Sample Traps section:

Collapsing Roof

This trap uses a trip wire... A successful DC 15 Dexterity check using thieves' tools disables the trip wire harmlessly. A character without thieves' tools can attempt this check with disadvantage using any edged weapon or edged tool. On a failed check, the trap triggers.

-DMG p.122

It's probably allowable to use improvised Thieves' Tools (in most situations) providing you have something sensible to make them from (at your DM's discretion). In addition, any lock picking or trap disarming check made with improvised Thieves' Tools should be made with disadvantage (if it's allowed at all).

Part 2: Do you need to be proficient with Thieves' Tools to use them?

This is the trickier part of the question as there are a few contradictions and unclear rules for this in the PHB/DMG. An important thing to note though, is that there isn't a section in the PHB/DMG that handles lock picking or trap disarming specifically. This means that a lot of it comes down to you and your DM's interpretation of any references to using Thieves' Tools.

There are a good number of references to Thieves' Tools in the PHB and DMG so let's look at the relevant ones and see what they say:

Equipment

The item descriptions for the Lock and Manacles both reference Thieves' Tools (though interestingly the Hunting Trap doesn't).

Lock. [...] A key is provided with the lock. Without the key, a creature proficient with thieves’ tools can pick this lock with a successful DC 15 Dexterity check.

-PHB p.152/PBR p.49

and

Manacles. [...] Each set of manacles comes with one key. Without the key, a creature proficient with thieves’ tools can pick the manacles’ lock with a successful DC 15 Dexterity check.

-PHB p.152/PBR p.50

Both of these listings could be interpreted to say you have to have proficiency to be able to pick the lock. However, as Cthos pointed out, a (fairly strict) reading doesn't actually exclude anyone who isn't proficient from making the check.

This could mean that a creature without proficiency in Thieves' Tools faces a DC 15 Dex check, a DC 15 Dex check with disadvantage, or even just a higher DC Dex check instead.

Tools

The general Tool rules and the Thieves' Tools description both point towards proficiency being non-mandatory:

Tools

[...] Proficiency with a tool allows you to add your proficiency bonus to any ability check you make using that tool.

-PHB p.154/PBR p.50/DMG p.239

Note: The DMG wording is infinitesimally different (and essentially identical) to the rules in the PHB/PBR

Since the phrasing of this is non-exclusive, like the equipment listings, it implies that you can take a non-proficient check with any tool.

Effectively, adding a proficiency bonus to a check (or not) is meant to simulate the "real world" impact of being proficient (or not) with whatever is involved in the ability check you're making.

Thieves' Tools. [...] Proficiency with these tools lets you add your proficiency bonus to any ability checks you make to disarm traps or open locks.

-PHB p.154/PBR v2 p.51

This description gives absolutely no indication that you need to be proficient to use Thieves' Tools, which I think is one of the strongest indicators that proficiency isn't required.

This is (quite surprisingly) the only main point of reference across the rule books. In addition, it also indirectly implies you can make a non-proficient check to disarm traps or open locks without getting your proficiency bonus and without disadvantage.

Ability Checks

Working Together Sometimes two or more characters team up to attempt a task. [...] A character can only provide help if the task is one that he or she could attempt alone. For example, trying to open a lock requires proficiency with thieves’ tools, so a character who lacks that proficiency can’t help another character in that task.

-PHB p.175/PBR v2 p.59

This is the second strongest case for proficient use only - after all, it is explicitly stated that you need proficiency to open a lock. This is the only reference to outright state this though and it does contradict the implications of all previous rules.

It could be that in this case "specific beats general" and Thieves' Tools specifically need proficiency whereas other tools generally don't. However, this would make them the only tools to need proficiency.

I also think this could very well be errata, missed in the final edits, based purely on where it is (i.e. this isn't in the description for Thieves' Tools).

Other Dexterity Checks. The DM might call for a Dexterity check when you try to accomplish tasks like the following:

• Control a heavily laden cart on a steep descent
• Steer a chariot around a tight turn
• Pick a lock
• Disarm a trap
• etc...

-PHB p.177/PBR p.60 (v3)

Again, this is a reference to taking a Dex check with no other qualifiers requiring proficiency (this was pointed out by Jason K in the question)

Mapping a Dungeon

Locked Doors

Characters who don't have the key for a locked door can pick the lock with a successful Dexterity check (doing so requires thieves' tools and proficiency in their use).

-DMG p.103

Ok, this provides the strongest argument for needing proficiency to use Thieves' Tools (and it's a strong argument one at that). This quote is pretty unflinching in terms of RAW and could be a deal breaker. However, as Jason K has pointed out, it is in a section that isn't about tools so could potentially be errata (and it isn't backed up by any other statements in the DMG.

Traps

If the adventurers detect a trap before triggering it, they might be able to disarm it, either permanently or long enough to move past it. You might call for an Intelligence (Investigation) check for a character to deduce what needs to be done, followed by a Dexterity check using thieves' tools to perform the necessary sabotage.

-DMG p.121

Here, again, is another reference to making a Dexterity check without explicitly requiring proficiency. This could have just been left out if it's already assumed you do require proficiency but that's not really stated as a normal assumption.

Twitter

Finally, for completeness, here's the tweet and reply between WaxEagle and Mike Mearls (co-lead designer of D&D 5e).

Important: I have stolen this directly from WaxEagles's answer purely for completeness. The effort on WaxEagle's part should be acknowledged here and I take no credit for this quote.

Conversation

WaxEagle: Could I get some clarity on thieves tools and lockpicking. Do you need the tools? Do you need proficiency?

Mike Mearls: anyone can use tools, prof bonus adds if proficient. No tools, DM can say no check or disadvantage.

Again, this is a pretty major point for allowing non-proficient lockpicking (or trap disarming) but I believe Mike has previously stated that Jeremy Crawford is the "rules guy" so it's not absolute (unfortunately).

Conclusion

After reading into the PHB and DMG, finding these rulings and taking them all together, my final assessment is still the same: I'd suggest that we can (and should) make the assumption that characters can attempt to use any tool they want.

Ultimately there is nothing physically preventing any character from picking up any tool, weapon, musical instrument, or set of armour they aren't proficient with and then trying to use it.

That said, a non-proficient character doesn't get their proficiency bonus and may also take an additional penalty, or have disadvantage (see non-proficiency with armour), so they're definitely less likely to achieve the outcome they want if they do try something unfamiliar.

If you're going for a "Gritty Realism" theme (DMG p. 267) to your campaign, then yeah, you could (house)rule that you need to be proficient and no amount of Dexterity is going to help without some kind of training or practice.

And that does kind of hold up in real life - ever tried to pick a lock before? However, I think a lot of people would agree that these guys, your characters, are heroes! I'd prefer to give them the benefit of the doubt, at least where RAW is concerned.

It would be great to see a definitive answer from the development team of 5e (Jeremy Crawford, Mike Mearls, & co.) but I'm not sure how likely that is. We shall see...

To restate my conclusion:

You do need Thieves' Tools to pick a lock but you don't need to be proficient with them.

Epilogue: How this works in practice

If you're still here and agree with my assessment, all this effectively means there are really only four scenarios you would come across when picking a lock:

1. You have Thieves' Tools and are proficient with them. You can pick the lock and get to add your proficiency bonus to the (Dex) check.

2. You have Thieves' Tools but you aren't proficient with them. You can still pick the lock but you don't get to add your proficiency bonus (since it's a bonus you only get when you are proficient with something).

3. You don't have any Thieves' Tools but you improvise some (with your DM's approval). You can still pick the lock but it's probably with disadvantage. If you have proficiency with the tools, add your bonus. If not, good luck with your roll!

4. No Thieves' Tools and no improvised tools. Take a strength check to throw the closest party member through the door or crowbar the lock. Basically, look for another way to get past it because you can't pick it.