Specify your needs
Most of cases we are looking for general purpose items like transistors, resistors, diodes and that sort of stuff. Always start looking at the cheapest alternative you can get. For resistors you can play with the tolerance if it does not has to be dead on you can go for +/-20% (pull-up, current limiting resistors). There is a huge selection of general purpose transistors it is always good to keep them stacked in your storage boxes and search for them regularly and try to find the best possible price (100 pcs / £1 is fair enough). Diodes tend to have many alternatives you can save some bucks with a few minutes google search. Of course if you need a specific part for a repair or someone else's project you might wanna look at them in bulk. Buying 10-25-50 sometimes cheaper than than buying 2.
If you are going to get parts via auctions you should go way under the buyout price and bid on as many items as you can pay for in case of winning them all. Look for new sellers for lower prices and fewer bidders.
Well, personally I don't like making offers. It's hard to find the middle path where you can save the most and the seller is still going to say ok. I you go too low he/she is going to discount only a couple percent. Of course you can be shy as well and blame yourself why did not you typed in a lower price. Make offers only if you are ok by paying the original price.
If you are looking for developer tools, Arduino boards or any other trending stuff you should definitely snipe. Auction sniping, is the practice of placing a higher bid at the last possible moment (often seconds before the end of the auction), giving the other bidders no time to outbid the sniper. This can be done manually, or by software. There is many softwares out there but none of them beats the old manual 4-10 browser windows method.