If you are a level 5 fighter who just picked up two-weapon fighting, even if you don't have multiple attacks, would you be able to attack with two weapons?
Specifics: Weapon training: blades, heavy; using a pair of Longswords +1
Yes. In Pathfinder, any character can fight with two weapons by taking a full-attack action to attack with his primary weapon and his off-hand weapon. Having a higher base attack bonus merely gives you extra attacks with your primary weapon.
From the the combat section of the Pathfinder System Reference Document, here is the relevant text:
Longswords aren't light, so at level 5, you could use a full-attack action to make one attack with your primary longsword at a -4 penalty, then one attack with your off-hand longsword at a -4 penalty. At level 6, you would be able to make two attacks with your primary longsword–one at a -4 penalty, and one at a -9 penalty–then one attack with your offhand weapon at a -4 penalty.
I'm no expert on optimization, but I suspect you would be better off using a light weapon in your off-hand. That would reduce the penalties for each attack by 2 and have little effect on your damage.
Note that using two-weapon fighting requires a full-attack action, not a standard action. In general, this means you can't move and attack with both weapons in the same round. However, you can use a standard action to attack with either weapon.
Fighting with two-weapons automatically grants an additional attack, which is in addition to the number of attacks one is granted from their Base Attack Bonus. A character who normally has 1 attack will have 2 when fighting with two weapons under normal conditions, and likewise someone with 4 attacks would have 5 (however, it's likely to be more - feats revolving around such a build continue to add an even greater number of attacks.) One can decide to fight with "two weapons" even when unarmed, leveraging the use of both hands (or simply twice as much abstract body-parts, if a monk), meaning that anyone can simply gain this extra attack whenever they wish (although as with all cases of multiple attacks, it's only useful when taking a full-round action), and even a commoner can fight with two attacks per round (although not very well - the Two-Weapon Fighting feat is critical to make it viable).