Early in the game, it's the Vault Suit. This is primarily for the radiation resistance, but it also has good energy resistance. Both of these things can be upgraded at an armor workbench.
If you're unconcerned with radiation, then you might pick another piece of clothing based on the buffs to your SPECIAL stats. This will depend on your character build and the stats that matter to you. Any of the items listed in Nukapedia's Clothing table will allow you to put on your armor without restriction. Some of the items in the Outfits table will allow armor as well, but not all the pieces.
For gun users, the Minuteman Outfit is good, as it offers Perception and Agility. Another option would be the Military Fatigues, which offer both +2 to Agility and 5 Energy Resistance. I often use these when I'm inside and not worried about radiation.
For Melee users, you might want to add to your Strength, perhaps with Army Fatigues, which add +1 to Strength and Agility. Another good melee option is the Grognak Costume. This is technically an outfit, as it doesn't allow you to wear a chest piece. However, it includes some good inherent damage resistance, +2 to Strength, and a great 20% bonus to melee damage.
Later in the game, if you acquire the ability to add Ballistic Weave to clothing, the situation changes completely. At this point, you can use an armor workbench to add a huge amount of damage resistance to clothing and outfits. From 30 each of damage and energy resistance with no ranks in Armorer, to 110 of each with Armorer IV. Notably, you cannot add Ballistic Weave to the Vault Suit. You'll likely want to choose the piece of clothing that buffs the stats you need most, and then add the highest Ballistic Weave you can create.
Note that not all clothing can have Ballistic Weave added. There's a list of possible options on the Ballistic Weave page I linked. Most of them are actually outfits. The only clothing included, meaning that you can equip all armor slots, are these:
- Army Fatigues (+1 Str and Agi)
- Baseball Uniform (+1 Str and Agi)
- Dirty Army Fatigues (+1 Str and Agi)
- Green Shirt and Combat Boots (+1 End and Cha)
- Military Fatigues (+2 Agi and 5 Energy Resist)
- Minuteman Outfit (+1 Agi and Per)
- Tattered Rags (+1 Luck)
You can also add Ballistic Weave to a few hats:
- Battered Fedora (+1 Luck)
- Green Rag Hat (10 Energy Resist)
- Trilby Hat (+1 Cha)
- Newsboy Cap (+1 Cha)
As you can see, none of these have any radiation resistance. So, if you wear ballistic weaving clothing and a hat, you'll most likely want to add Lead lining to at least one piece of armor. I usually add it to each arm, and that seems to be sufficient.
It's also useful to carry additional pieces of clothing for use when not in combat. I carry both Charisma and Intelligence gear with me at all times, and even hotkey it. You can often access clothing from your hotkeys in the middle of a conversation.
You can easily find hats that add +1 to Charisma and Intelligence (Newsboy Cap on the guy that tries to kill his brother in the Diamond City Market, Press Cap on Piper).
You can find clothing that adds +2 to Charisma pretty easily as well. There are many types that do this, including most suits and dresses. You can get +3 to Charisma clothes by completing Curtain Call. Finding +2 to Intelligence clothing such as a lab coat usually takes longer, but should happen by the time you're in your teens. I've found a lab coat in the first suitcase in Sanctuary, though, so results vary.
You can find +1 to Charisma glasses, including Black-Rim Glasses and Fashionable Glasses, pretty easily. There exist glasses that give +1 to Intelligence, the Road Goggles, but these are more rare. I got some from the Pillars of the Community. If you have the Automatron DLC, then you can get a pair from Zoe's corpse. There are also +2 to Int (and -1 to Cha) glasses called Liam's Glasses. To get these, you must complete Plugging a Leak while siding with Liam. Note that this quest is unavailable if you've completed Tradecraft, which means that you can't get both Liam's Glasses and Ballistic Weave. Choose the weave. If you have the Far Harbor DLC, you can get The Captain's Hat, which gives +2 Int.
There is also a legendary armor mod, Sharp, which adds +1 to Int and Cha. If you have one or more pieces of this on, you might not need to swap items.
When you're trading or persuading, put on the Charisma gear. When you're doing noncombat things that give you experience, like crafting, picking locks, or turning in quests, put on the Intelligence gear. Putting all your armor back on after switching your main clothing is a bit of a hassle, so sometimes I'll just swap the glasses and hat.
Remember that clothing is irrelevant while wearing power armor, as the only equipment that affects your stats at that point is the armor itself.