Moving Servers - Help? :S

[Gallio]

Hello, I'm trying to move servers again I have problems everytime is seems I tried restoring via PHPmyAdmin but I get a timeout, I'm really looking for some help I don't quite understand SSH I do have access but then again it doesn't matter because I have no clue in how to use it. I really need some help here, If anyone could give any good tips on moving servers fast and simple. Keep in mind I have no knowledge for SQL at all. I'll give you my server specs below.

First Server (The one I'm moving from)

Operating system Linux
Service Status Kernel version 2.6.9-22.ELMachine Typei686
Apache version1.3.34 (Unix)
PERL version5.8.7
PHP version5.0.4
MySQL version4.1.13-standardcPanel

Second Server (The one I'm moving to)

Operating system Linux
Kernel version 2.6.9-22.0.1.ELsmp
Apache version 1.3.34 (Unix)
PERL version 5.8.5
PHP version 4.4.1
MySQL version 4.1.13-standard

Hopefully someone can help -Dylan