Does being connected to more peers relay transactions faster? If I'm connected to 500 peers, will I know about transactions faster than if I'm only connected to 8 peers?
No, very little of the time for downloading new blocks is actually related to network transmission.

Most of the work being done is the I/O against your hard drive as the program writes data to the BDB database storage.  If you are on an encrypted filesystem, the problem is exacerbated.  There is some CPU work as the elliptic curve encryption is used when verifying the signature for each transaction so a faster CPU will help as well.

Some people use SSD disk storage and report quite respectable download times 

Some improvements are coming in v0.7 of the Bitcoin.org client but the real signficant performance improvement will come from the use of LevelDB and "ultraprune", both of which are expected in v0.8.