First of all, you need to get the last version of Safari that is compatible with OS 10.6.8, version 5.1.10. You can get that from here:
http://www.macupdate.com/app/mac/15675/apple-safari
Don't click that first download button you see! That will download version 8.0.3, which won't work with 10.6.8. About 1/3 of the way down, you'll see
Download Safari 5.1.10 (OS X 10.6.8). When you click on that right there (that is, click on where it says "Download Safari 5.1.10 (OS X 10.6.8)"; it's already a hyperlink), you'll get a "dialog box" to download it, click on "Save File", and it will download Version 5.1.10.
Same with Chrome. Here is the link to get the latest version:
http://www.macupdate.com/app/mac/32956/google-chrome
Now, here are the google results for a search I just did for "On a Mac, DNS lookup failed message":
ttps://
www.google.com/search?q=On+a+Mac%2C+DNS+lookup+failed+message&ie=utf-8&oe=utf-8
Not sure which one will help you directly, but a number of them mention the "DNS Server" address. Not sure where you are located (I am near Seattle, Washington, in the USA), but on my MacBook Air, here is what is listed for the various fields used for Network connections:
Via the Network preference pane, I first get "Connected" for the Status. Clicking on Advanced brings up a page that has 7 "headings" (Wi-Fi, TCP/IP, DNS, WINS, 802.1X, Proxies, and Hardware). The Wi-Fi one is already selected, and in the box shows the Network Name, and the Security "protocol" (mine says WAP2 Personal"). When I click TCP/IP, it shows the following:
Configure IPv4: Using DHCP
IPv4Address: 192.168.1.5
Subnet Mask: 255.255.255.0
Router: 192.168.1.1
Clicking on DNS shows the DNS Servers: as 192.168.1.1 (same address is the IPv4Address, and the Router address (I have a Netgear Router, and I use a "special" Arris TM 722g DOCSIS 3.0 Telephony Modem, as I have both Internet and Landline phone services through my ISP, Comcast)).
Quite a few of those links talk about maybe changing the DNS Server address. You will need to read through some of those, as I don't have a simple answer for you. Maybe someone else with more intimate knowledge about this can help in a more definitive way, and especially with where you are located.
Good luck!