Small Guns

In addition, to increase this skill, you can use 10 "Guns and Bullets" books. Rodrigues in Den can increase your skill +5% if you have skill between 81 and 119 for some money.
Can be increased by 10 after finishing quest from Valerie in VC.

Big Guns

Marcus in BH can increase this skill +5% after completing the quest.

Energy Weapons

In addition, to increase this skill, you can use 10 "Cat`s Paw #5" magazines. They are pretty rare, can be obtained from special enc from a very fat man.

Unarmed

Joe Sullivan can improve your HtH skill +30% after finishing his quest.

Melee weapons

Walk through "Replication #13" location to improve this skill. Sally to 40% (if it`s less) and Zang to 60-69%

Throwing

First Aid

Recovering HP goes by this formula: HP=[Skill/(11-Luck)]
When having "Field medic first aid kit" in an additional Slot: HP=[(Skill*10)/(100/Luck)]
Cooldown = 7500/Skill
First-aid Kits do not affect the Cooldown.

In addition, to increase this skill, you can use 10 "First Aid" books. You can take Medic perk on 30-32 lvl.
Can be increased by "Master of Medikit" perk.

Doctor

Cooldown = 18000/Skill
If you are bleeding or you are crippled, the usage of doctor removes this effect, but only that (roll on healing limbs doesn't work).
If the doctor skill is 150 or above, and your health is lower than a fourth, there is a calculation for the healing of HP (about a third of the doctor skill). As well as the removing of radiation and poison.

When healing limbs (blindness) calculation is the following.
Skill. To this will be added +50("Paramedics bag"), +25 (Doctors Bag), +25 ("Field medic first aid kit"), +10 ("First-aid kit").
Bonus to roll +25("Paramedics bag"), +10 (Doctors Bag).

Chance = Skill*6/10.
If 1 eye is damaged: Chance/2, if both: Chance/4. With one eye knocked out, the chance is cut by half, And with both there is just a quarter of the chance left.

Next, we check each limb and roll: (Chance+Bonus to roll) if more than (Random(5,95)) - the roll is successful and the limb is treated.

Example. Having 100 in Doctor and lost 1 eye, with a "Field medic first aid kit" in your slot you get:
125*6/10 = 75, 75/4=18 (rounding to the next lowest).
Somewhere about 20% to cure the eyes.

The same, but with "Paramedics bag"
150*6/10=90. Bonus to chance +25. 90/4=22. 22+25(Bonus)=47.
Somewhere about 50% to cure the eyes.

This is how it works currently.
To increase Doc skill you can take Medic perk on 30-32 lvl.
Can be increased by "Master of Medikit" perk.

Sneak

Entering Stealth is only possible when having 1 out of the following 3 things: Stealthboy, "Scout costume", "Special armor".

Lockpick

You can increase Lockpick skill after compliting a chain of quests in Arroyo. 200 lockpick skill needs for +1AP quest (Quest from Rianna in Den, if you`ll have no of Lockpick skill - you not obtain +1AP)

Pickpocket

In addition, to increase this skill, you can use 20 "Pre-war" books. Chance to increase skill by book = 33.3%
Don`t try to use less then 300 skill to steal from player or NPC, it`s big chance to fail.

Traps

Jordan in Arroyo can increase your skill on 5% as a reward of his quest.

Science

In addition, to increase this skill, you can use 10 "Science" books. Can be increased by "Master Crafter" perk.

Repair

In addition, to increase this skill, you can use 10 "Repair" books. Can be increased by "Master Crafter" perk. Can be increased by "Master of Repair" perk.
Can be increased after completing a chain of quests from Joe in Modoc. Can be increased by 10 after finishing quest from Valerie in VC.

Speech

In addition, to increase this skill, you can use 20 "Pre-war" books. Chance to increase skill by book = 33.3%. Can be increased by perks after softcap +40% in total.
Gives you a chance of quests simplification, ability to check the location of needed player via hubologist in a bank.

Trade

For now the best way to increase Barter skill - to increase Barter Exp instead of it. This skill is rise while you trading, and increases Barter skill as well.
So you need to find a trader with shitty items (caravans or traders from Klamath, Den etc) and trade with them until your barter skill will be on the same level as theirs.
Merchant`s barter skill For example: if you will trade with Bob in Klamath (he has 30 Barter skill) you will have some penalty in trading. But when you will be near 30 limit - you will have cheaper prices.
At the point of 30 Barter skill, you will have the best price, but can`t increase your barter skill anymore with this trader. So you need to find someone with 60-70 skill.
Of course, you can find a trader with 300 barter from start…. But prices will be fantastic in this case.

Athleticism

Formula: Walk = 400-Skill/3; Run = 200-Skill/10.¹

Outdoorsman

In addition, to increase this skill, you can use 10 "Outdoorsman" books. From start you can increase this skill to 60-69% (depends from initial value) from Smiley in Klamath.
Slim in Klamath can increase your skill 2% for 100 coins. Kameron in Arroyo +1% of skill after finishing his quests.