Go to a website like redfin.com or zillow.com where you can see all of the previous sales in the neighborhood. You might discover that houses that actually sold went for much less money than the houses that are on the market. Tour other houses in the neighborhood. Talk to agents openly about neighborhood prices (not just your own agent, but listing agents as well---of course, the information they give you will be biased, but they will point out useful things you weren't aware of). Talk to locals in the neighborhood. After a few weeks of research, you should have you a pretty good sense of what the "fair market value" of the home is.
In fact, you may have a better idea than the home seller does, especially if the market has been declining and the seller doesn't realize this (or is in denial). Because of this, even if you make a fair or slightly below fair market value offer, there is a risk the seller will brush you off or make an unreasonably high counteroffer. Don't let that bother you. Don't let your agent pressure you into offering an unreasonably high number. Just be patient and keep making serious offers until you find somebody you can work with.