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Bardic Performance reads as follows:

Starting a bardic performance is a standard action, but it can be
maintained each round as a free action. Changing a bardic performance
from one effect to another requires the bard to stop the previous
performance and start a new one as a standard action. A bardic
performance cannot be disrupted, but it ends immediately if the bard
is killed, paralyzed, stunned, knocked unconscious, or otherwise
prevented from taking a free action to maintain it each round. A bard
cannot have more than one bardic performance in effect at one time.

At 7th level, a bard can start a bardic performance as a move action
instead of a standard action. At 13th level, a bard can start a bardic
performance as a swift action.

I'm going to play a Sound Striker and my plan is to use Weird Words an offensive bardic performance as a standard action and then use Inspire Courage to buff my party as a movement action.

Is that possible? Is there any rule that prevents me from starting more than 1 bardic performance per round?

7th level bards

All bards, from 7th level on, can start a new performance as a Move Action. From then on, you can either use a standard or a move action to start a performance, and thus start two different performances in a single round.

At 7th level, a bard can start a bardic performance as a move action instead of a standard action. At 13th level, a bard can start a bardic performance as a swift action.

At 13th, you should be able to start up to three different performances in a round, using your standard, move and swift actions.

Do note that you can only have one bardic performance in effect at any given time. So, normally, you couldn't have Inspire Courage and Inspire Competence active simultaneously, as you have to stop one performance before you start another.

A bard cannot have more than one bardic performance in effect at one time.

However, you are not in fact using two performances simultaneously, as the Sound Striker's Weird Words ability is not a bardic performance that lasts for a number of rounds. It is a supernatural ability that "consumes" your daily uses of bardic performance, create a magical effect, and dissipates. In other words, the Weird Word performance is like an instantaneous effect, it begins and ends immediately.

Just remember that Weird Words is "always a standard action" to use.

Mythic Bards

Mythic bard characters who take the Marshal mythic path may learn the Lightning Performance ability, which allows them to start a new performance as a Swift Action instead of a Standard Action:

Lightning Performance (Ex): Your tongue is as quick as any blade. You can start a bardic performance as a swift action instead of a standard action. If you expend one use of mythic power, you can instead start a bardic performance as a free action.

Using this ability would allow you to start two performances in the same round before you are 7th level, one using your standard action and another using your swift action, as mythic paths follow a different progression from normal character advancement. Meaning that you could be a first level bard with a mythic tier 1 and this ability to start performances as a swift action.