Fallout: New Vegas skills

Background
Fallout: New Vegas shares a similar skill set to its predecessor, Fallout 3, but with several differences. The primary weapons skills have been revamped, with Small Guns and Big Guns having been combined into the Guns skill, which now covers all conventional firearms. Weapons using explosive ordinance (missiles and grenades, for instance) are now covered by Explosives. New throwing weapons, such as the throwing spear, have been added, and are categorized as Melee Weapons. The Survival skill has also been added to the roster. It allows for crafting of various items, such as food, stimpaks, and poisons to augment the effectiveness of weapons.

Unlike Fallout 3, the base game will not give the player enough points to max out every skill, even accounting for every perk and skill book. It is possible to achieve a functional maximum in roughly ten skills by using skill magazines and the Comprehension perk for a temporary boost to 100 from a base level of 80. However, with two or more DLCs installed, 10 Intellingence, the Educated perk, and having consumed roughly 20 skill books, it is entirely possible to max out skill points by level 40.

To offset the necessity of a minimum value for a certain skill, New Vegas introduces skill magazines, which provide a large but temporary boost to their associated skill (+10 or 20 with Comprehension). By using one prior to conversation, a check can be passed that might otherwise be failed. They can also be used to boost skill with weapons temporarily or gain access to crafting recipes beyond the player's current skill. Like skill books, magazines are consumed once used, and more need to be collected to maintain the effect.

Effect of skills
Skills can be largely thought of as either "combat" or "non-combat": "combat" meaning skills that influence the effectiveness of weapons (also which skills Good Natured penalizes). Combat skills would be Energy Weapons, Explosives, Guns, Melee Weapons, and Unarmed.

Combat skills
For all combat skills, having a higher score helps meet minimum weapon requirements (some weapons also have a Strength requirement). Not meeting the requirements for ranged weapons drastically penalizes your V.A.T.S accuracy and increases weapon wobble. For melee/unarmed, you instead have a slower rate of attack.

For non-Unarmed weapons, your skill score also increases the damage done, by the following equation (where base damage is the damage listed on an individual weapon page):
 * $$Final\ Damage=Base\ Damage\times\Bigg(\frac{100+Skill}{200}\Bigg)$$

In other words, at a hypothetical score of 0, a weapon will only do half the listed base damage. At 100, the weapon will do twice that damage (or the actual listed base damage).

Unarmed skill does not affect unarmed weapon damage in this way. It does contribute to a bonus to unarmed damage, though (see the unarmed damage page for specifics). Any further increase in unarmed weapon damage is driven primarily by special unarmed attacks and by special V.A.T.S moves, which are unlocked at higher skill scores.

Note that your Pip-Boy 3000 will occasionally show the wrong listed damage, generally when you have a weapon selected but not equipped. (An example of this is a Fat Man when you don't have normal Mini nuke ammo.) Equipping the weapon will fix this issue.

Non-combat skills
Non-combat skills have varying specific uses. See individual skill pages for more details. In addition to their normal uses, however, many non-combat skills tend to be used as skill checks during dialogue (the main purpose of Speech in fact). Of note, Barter tends to be used as a back-up to Speech skill checks in some conversations. Combat skills do occasionally get used in dialogue skill checks, though not nearly as frequently.

Moreover, unlike past Fallout games, New Vegas uses the idea of uncontested rolls; that is, instead of a probability of success based on your skill score and the difficulty of the check, there is a minimum threshold to meet, which results in either an uncontestable success or failure.

For example, in order to convince Easy Pete to provide some dynamite to protect Goodsprings during the quest Ghost Town Gunfight, the player must have a minimum Explosives skill of 25 (an example of a less-frequent combat skill check in dialogue). In the case where a player's associated skill level is too low, a dialogue option (highlighted in red) is presented that will fail the skill check, and will not grant a speech success. Unlike Fallout 3, where the same dialogue option is presented regardless of your success or failure, a check that will fail uses a humorously unconvincing response, while a passable check uses a well-thought out argument, thus reflecting the nature of the check. XP is awarded in proportion to the difficulty of the check.

Formula
The initial value of each skill is a base value of two, plus an amount depending on a character's value in the relevant attribute, plus a bonus determined by their Luck attribute, rounded up.
 * $$2 + (2 \times \mbox{Stat}) + \left\lceil\frac{\mbox{Luck}}{2}\right\rceil$$

Example: A starting Endurance of five and a starting Luck of five will give you an initial Unarmed skill of 15.
 * $$2 + (2 \times 5) + \left\lceil\frac{5}{2}\right\rceil = 15$$

Later changes to the SPECIAL stat have a similar influence on the respective skill.

During character creation, the player will tag three skills, instantly adding 15 points to each one. When leveling up, the character will distribute ten skill points plus a number equal to half their Intelligence, totaling 15 at maximum IN. The Educated perk grants two additional skill points per level if chosen. Assuming the player has an Intelligence of ten from level 1 and takes the Educated perk at level 4, a maximum of 487 skill points can be distributed without any DLC. Each DLC raises the level cap by 5, thus adding a possible maximum of 5 x 15 = 75 points (or 5 x 17 = 85 points with Educated) to the total available.

Skill Points available with a starting INT of 10 and with / without Educated:

Base game: 487 / 435

One DLC: 572 / 510

Two DLC: 657 / 585

Three DLC: 742 / 660

Four DLC: 827 / 735

It can be seen from the above figures that the Educated perk confers an advantage relative to the number of DLCs / levels, so that a player with three DLCs and Educated actually has more skill points to distribute than a player with four DLCs but without Educated.

Maxed skills
With all DLCs, it becomes fairly trivial to max out all your skills at 100.

At the base level, you need 1300 skill points for complete maximization (100 points for 13 skills). At level 1, you get 45 skill points for free (from your 3 Tag! skills), and an average character (5 in all SPECIAL stats) will have 15 as a base for all skills. This reduces the amount of skill points you need to 1060 (1300 - (45 + 15 x 13)).

For 49 levels at average intelligence, you get 710 skill points (14.5 x 49). Just with these base stats, this is enough to get all but 5 skills completely maxed out (remaining 350 / 85, rounding up).

Simply by taking Educated at level 4 leaves us with 258 skill points remaining to completely max out. Taking Comprehension at level 6 (and saving skill books until then) means that reading 65 skill books is enough to net you the remaining skill points (so long as you adapt your per-level skill points to the remaining skill books). As a note, the base game contains 53 skill books and Old World blues contains 16 skill book recipes, which is already more than enough to meet this (Dead Money also contains 12 and Honest Hearts can contain anywhere from 0 to 8, 0 to 2 per workbench).

With simple variations, the task of maxing out skills can become easier or harder. A character with Intelligence of 10 with Educated at level 4 and starting off with Skilled will be short only 408 skill points, not including skill points from initial character creation (which will yield at least 111: 45 from three Tag! skills and 22 from Repair/Medicine/Science, ignoring Luck). This character will need only very few skill books by comparison, about 42 if all other SPECIAL stats are left at 5.

All of the above also ignores the effect of implants from the New Vegas medical clinic, the special Old World Blues perks, and other perks and traits that can effectively make permanent alterations to your SPECIAL stats, which can be as much as 13 extra skill points (increasing Luck by 1 to an odd number) and at worst 2 (Strength only affects Melee Weapons).

In short, due to the increased level cap, extra content, and perks/traits from DLCs, the task of maxing out your skills is nowhere near as exacting as it once was. Even "dumb" characters (Intelligence of 1 and no special perks or traits) will be able to max out many skills.