General movement:
* Approached
* Moved toward
* Headed for
* Progressed towards
* Made for
* Strove towards
* Tended towards
* Advanced towards
More specific movement:
* Marched towards
* Crawled towards
* Ran towards
* Floated towards
* Soared towards
* Glided towards
Figurative language:
* gravitated towards
* inclined towards
* steered towards
* aspired to
* sought out
Formal:
* Proceeded towards
* Directed towards
The best word to use will depend on the specific situation and the tone you want to create.