You’re missing the two-weapon fighting combat option rules. Without those, the feat et al. won’t make very much sense.
The two-weapon fighting combat option allows you to get an extra attack when you full attack, but all of your attacks in that full attack take a penalty. The extra attack has to be with a different weapon from your regular attack (“offhand” weapon), and the penalties you take depend on whether or not that offhand weapon is light or not.
The Two-Weapon Fighting feat reduces those penalties to a point that this option is actually usable.
Another key thing to realize is that having two weapons in hand does not automatically mean you are using the two-weapon fighting option. You have to choose to use it, to get an extra attack and take an attack penalty.
As for the +6/+1 thing, that’s called an iterative attack, and that’s free just for having BAB that high. It is separate from, and can be combined with, the two-weapon fighting option. So, assuming you have the Two-Weapon Fighting feat and a light weapon in the offhand (the only case in which the two-weapon fighting option is a good idea), with +6/+1 BAB you could attack normally, for +6/+1, or you could choose to use two-weapon fighting, taking a −2 penalty. Then your attacks would be +4/+4/−1, where the second +4 attack has to be with the offhand weapon. If you take the Improved Two-Weapon Fighting feat, that becomes +4/+4/−1/−1, because Improved Two-Weapon Fighting gives you a second bonus attack with the offhand weapon, at −5 (matching the penalty on the iterative attack).
Pathfinder: You can swap between weapons during a full-attack
You can make a full-attack and use any weapons available to your character for each of your attacks. This has a lenghty FAQ item explaining what is possible during a full-attack and it's worth reading for full understanding of the rules for fighting with two-weapons.
Multiple Weapons, Extra Attacks, and Two-Weapon Fighting: If I have extra attacks from a high BAB, can I make attacks with different weapons and not incur a two-weapon fighting penalty?
Yes. Basically, you only incur TWF penalties if you are trying to get an extra attack per round.
Let's assume you're a 6th-level fighter (BAB +6/+1) holding a longsword in one hand and a light mace in the other. Your possible full attack combinations without using two-weapon fighting are:
(A) longsword at +6, longsword +1
(B) mace +6, mace +1
(C) longsword +6, mace +1
(D) mace +6, longsword +1
All of these combinations result in you making exactly two attacks, one at +6 and one at +1. You're not getting any extra attacks, therefore you're not using the two-weapon fighting rule, and therefore you're not taking any two-weapon fighting penalties.
This has been answered on this FAQ item about monks aswell.
But to put it simply, you only take two-weapon fighting penalties if you want to make extra attacks using the weapon of your off-hand.
This is only an invalid option if you are using both hands to wield a two-handed weapon and wish to use a weapon on your off-hand, like a spiked gauntlet or claw attack, as explained on this FAQ item.
D&D 3.5: You also can swap between weapons
This ruling seems to be exactly the same for d&d 3.5, because the rule for full attacks does not limit your character to a single weapon, and says you can use whichever want first when you make a full attack:
If you get multiple attacks because your base attack bonus is high enough, you must make the attacks in order from highest bonus to lowest. If you are using two weapons, you can strike with either weapon first. If you are using a double weapon, you can strike with either part of the weapon first.
However, your free off-hand attack must be from your off-hand weapon, since that is the condition given by the two-weapon fighting rules.
Ignore the Rules of the Game articles (3.5)
The Rules of the Game (from Skip Willams, one of the authors from the players handbook) had a series of three articles that discussed in detail how attacks, full attacks and two-weapon fighting works. Looking for clarification, i naturally stumbled upon them and have to mention on this answer, as they were written by official sources.
Being relative to this answer, i have to quote the first article on that series that says:
When fighting with two weapons, you gain one extra attack with your off-hand weapon when you use the full attack action. If you have a high base attack bonus, you gain iterative attacks only with your primary weapon.
His example on on the second article, about how main-hand weapon and off-hand weapon works, go even futher, and explain that off-hand penalties occur even when not using the full-attack action, as seen here:
If, after you made two-weapon attacks with your sword and torch, a foe later provokes an attack of opportunity from you that same round, you can strike that foe with your longsword with no two-weapon penalty at all. (You also can use just the torch, also with no two-weapon penalty, though you still take the -4 penalty for an off-hand attack; you also still take the -4 penalty for an improvised weapon for a total penalty of -8.)
You might be reading this and thinking wait, none of the rules says anything like that.
Yes, exactly, they don't.
These articles have been labeled as bad sources for rules by the community, as they were full of contradictions compared to the published books.
Best Answer
Yes, you can trip and attack in the same turn.
Trip, sunder, and disarm replace a single melee attack and thus can be interspersed in any order in multiple attack routines, as part of a flurry of blows, and as an attack of opportunity. This is clear from the descriptions of the maneuvers in the SRD.
Of course, these may provoke attacks of opportunity of their own, and other feats and weapons you may have might affect number of attacks and bonuses thereto. In our pirate campaign just today, we had a monk consistently tripping first in his flurries to benefit from the lowered AC of the prone opponent, and a big barbarian who would use his lower-to-hit second attack on disarms since he figured on the off chance he connects it might as well mess with the opponent more than just some damage. Make sure and read the prone rules and other related areas to understand how all this will work.