There are three basic answers to this, and it's entirely viable to mix and match them as best fits your character. The most important books to own, at least if you want to easily add this kind of feature to any character, are Complete Champion and/or Tome of Battle: Book of Nine Swords. Without these, you will have to dedicate a lot of resources (read: levels) to gaining these sorts of abilities.
Your three options are:
- Get a full-attack after/during another action that moves you, or without using a full-round action.
- Move without needing a move action
- Find good uses for standard actions so that you aren’t hurt so badly when you cannot full-attack
While I’ll try to break down your options for each below, the short, best answers to each of these is as follows:
- Pounce, from Barbarian 1 with the Spirit Lion Totem ACF (Complete Champion)
- Travel Devotion, taken as a feat or, better, gotten from Cleric 1 (Complete Champion)
- Martial Strikes, from any initiating class or the Martial Study feat (Tome of Battle)
Full-Attack Motion – Pounce
There are several ways to move, and then full-attack anyway. The most common and probably best method is Pounce, a special quality that allows you to make a full-attack at the end of a Charge, instead of only a single attack. Many feline monsters have this feature, for example.
The easiest way, bar none, to get Pounce is to take a single level of Barbarian, and, using the Lion Spirit Totem alternate class feature, printed in Complete Champion, trade Fast Movement for Pounce. Barbarian is a really solid level anyway, thanks to great HD, decent skills, and Rage. Note the Whirling Frenzy variant in Unearthed Arcana: it replaces Rage, and a Whirling Frenzy has a bonus attack as one of its benefits: this goes together very well with Pounce.
Other options include Lion’s Charge (Dru 3, Rng 2) spell (Spell Compendium) and Psionic Lion’s Charge (Psy 2) power (Expanded Psionics Handbook), pouncing charge maneuver (Tiger Claw 5, 1/enc w/o refresh, Tome of Battle), any type of shapechanging (e.g. Polymorph or Wild Shape) into a creature that has Pounce (most feline or feline-like monsters), and various prestige classes (e.g. Weretouched Master from Eberron Campaign Setting, Reachrunner from Races of Eberron, Sigh Rager from Oriental Adventures, Lion of Talisid from Book of Exalted Deeds, Geomancer from Complete Divine).
Alternatives to Pounce
Pounce is not the only way to move and full-attack, though it is the most common/least restrictive. These are some other options.
Dervish Dance: The Dervish prestige class (Complete Warrior) and the vestige Paimon, the Dancer (Tome of Magic) both give an unusual ability to make attacks as you move around. The rules are slightly different for each, but the gist of it is that you get to make an attack for every few squares you move, up to your usual maximum for a full-attack.
Shadow Pounce: This very rare ability only appears on a couple of Forgotten Realms prestige classes, the Crinti Shadow Marauder (Shining South) and Telflammar Shadowlord (Unapproachable East). Both have difficult prerequisites. The Shadow Pounce ability lets you make a full-attack following any teleportation effect. Due to the existence of teleportation effects that work on move actions and swift actions, this can mean three full-attacks per round, at least until those effects run out. Not many groups appreciate that, however.
Moving without using a move action almost always means moving with a swift action.
The best way to do this is with the Travel Devotion feat from Complete Champion: for one minute, you may move as a swift action, 1/day. You can use two uses of Turn/Rebuke Undead to activate the feat again. Taken on its own, this feat is pretty good, but most characters will have more than one fight in a day.
By taking a single level of Cleric, however, you can trade Travel Domain for Travel Devotion (see the rules for [Domain] feats in Complete Champion), getting the feat for free, and also get 3+Cha Turn/Rebuke Undead uses. If you have at least Charisma 12, this is enough for three separate combats, which will get you through most of the standard adventuring day. Consider the Undeath Domain, if you can find a way to combine Travel and Undeath thematically, for the free Extra Turning feat (enough to allow a Charisma 10 character to use Travel Devotion four times per day).
Note also that Cleric 1 also gives you some spells, and if you make it a Cloistered Cleric (Unearthed Arcana), you also get the Knowledge Domain (which you can trade for Knowledge Devotion, another excellent feat from Complete Champion), Lore, and really good skills. Cleric is the best single-level dip in the game.
Other options for swift action movement include the Anklet of Translocation (2/day, 1400 gp, Magic Item Compendium), sudden leap maneuver (Tiger Claw 1, 1/enc w/o refresh, requires Jump check, Tome of Battle), the dimension hop and hustle powers (Freedom 1, Complete Psionics, and Psy 2, Freedom 2, Expanded Psionics Handbook, respectively), and the flicker mystery (Shadowcaster 3, immediate action, Tome of Magic).
Better Standard Action Attacks
I’ll be ignoring the obvious option (casting a spell or similar) since this discussion is primarily about attacking.
Basically, this is about attacking on a standard action, and having it not be a huge hit to your damage output. The star here is Tome of Battle: Book of Nine Swords: initiators have many standard action Strikes that are quite potent. While initiators will do more damage with a full-attack, they have access to enough options that having to use a standard action is not going to mean their turn is wasted.
Note that non-initiator characters can access up to three maneuvers by taking Martial Study feats, and non-initiator class levels count as ½ Initiator Level for the purpose of taking them.
For ranged characters, there are Manyshot and Greater Manyshot. Manyshot is not a real full-attack, since it is only two arrows no matter your BAB and only applies precision damage once. Greater Manyshot still limits you to only two attacks, but at least they are separate attacks that get their own precision damage. Most ranged attackers should try to get bonus damage like this, so it’s important that it gets applied repeatedly. Then again, ranged attackers do not have to move as often.