It seems like the install DVD is unreadable for some reason, either the disk is bad (DVDs are less durable than you might think), or the drive is bad (have you attempted to read any other disks lately?).
In any event, the easiest way to get around it is to boot off your old hard drive, and clone it to your new one. You'll need a USB external closure that can take a 2.5" SATA drive (they can typically be had for around $20 or $30). Put your old drive in that, plug it in, and hold down option as you boot. You'll get a selector screen, and your old drive should be available to boot off of (it will likely be labelled Macintosh HD and have the orange external drive icon). Select that, and your Mac will boot just as it did previously. From here, there are a couple ways to get your new drive working.
Clone Your Old Drive
Unless you really want a fresh install for some reason, this is my recommended method. It will copy your entire system over to your new drive, so you'll get larger capacity, but without the hassle of re-installing everything.
To clone your drive, open up Disk Utility (found in
Applications/Utilities, or just use Spotlight). In the right hand pane, you should see your old boot drive, and your new drive. Select your old drive, click the Restore tab, set Source to your old drive, and Destination to your new drive. Then click Restore. This will likely take a few hours, but once you're done, you should be able to restart and boot into your new internal drive.
Fresh Install from USB
If you want to do a fresh install, you can try reading the installation disk to a USB drive, which may be more reliable. You'll need a sufficiently large USB hard drive or memory stick (for 10.5, you may need 8 GB, but it's possible 4 GB will do).
To start, insert the installation DVD and check that you can read it. If you can, then you're good to go, otherwise you'll need to get another copy elsewhere, or a newer version of OS X (or you can try the clone method).
The procedure to copy the installation disk is the same as cloning the hard drives, you just switch the Source to the installation disk, and Destination to your USB drive. Once the copy is complete, reboot and hold option, and select your USB installation disk from the boot menu. You should then see the standard OS X installer and you can proceed as normal.