I'm also new to mining and maybe can give you some hints.
First the prices of these cards are in 3 different regions:
5830 = ~ 90 €
5850 = ~ 115 €
6870 = ~ 150 €
I have a Sapphire 5850 Xtreme and a Sapphire 5830 Xtreme, both cards are nearly silent if you don't overclock.
Very good cards in terms of noise, overclocking ability, and price. Well designed, i am surprised.
With overclocking and a fan level of ~ 45% the cards are in my case not hearable, because the power supply is louder.
Card | stock clock | overclocked (Mhz @ fan-speed)
5830 | ~ 250 Mh/s | ~ 300 Mh/s (972 @ 45%)
5850 | ~ 275 Mh/s | ~ 350 Mh/s (900 @ 50%)
6870 | ~ 270 | ~ 320 (expected, from https://en.bitcoin.it/wiki/Mining_hardware_comparison
In case you just download a miner software like GUIminer http://forum.bitcoin.org/index.php?topic=3878.0
and fire it up, you dont get these values. In fact they would be much smaler.
Some tweaking is required, here are some steps:
1. with an ati-card i suggest to use phoenix miner http://forum.bitcoin.org/?topic=6458.0
(its integrated in guiminer), it gives me 10%+ more MH/s
° use this extra flags (in guiminer): -v -k phatk VECTORS BFI_INT FASTLOOPS=false WORKSIZE=256 AGGRESSION=10
° adjust AGGRESION to your needs, 10 and more if your pc is idle, when working on the desktop try 7 or 8. (It adjusts how much power goes into hashing, with higher numbers your desktop get lags, because phoenix is taking the power first to calculate and then at the second step, your Chrome window or office Doc gets minimized.
hope that explains it. You'll see some difference in the region of ~ 5 Mh/s.
2. downclock the memory to something like 300 Mhz, the advantage of this is a much more cooler card, which gpu can be easier and higher overclocked. I could not see any MH/s change with higher or lower memory clocks with phoenix. I think that is only important if you use poclbm or other miner-software.
3. for down/up-clocking use a tool like:
° Trixx http://www.sapphiretech.com/ssc/TriXX/
° MSI Afterburner http://event.msi.com/vga/afterburner/download.htm
° Catalyst Control Center ....
At stock core you will not get at 300 MHash/s with all of the cards. But don't worry, overclocking is very simple and until you don't push it up to high, it is very stable. Just give it a try.
i would suggest to buy a 58x0, price and hashpower is better compared to a 6870.