Negotiation is a skill, an art, perhaps obtained by common sense and experience.

As a contractor, I would not negotiate with you. I have a price, terms and conditions.

Are you requesting price reduction advice? If so, show us your statement of work/request for proposal.

Keep getting bids if you are looking for a lower price, but bear in mind that you often get what you pay for. Personally, I would never bother trying to negotiate with a contractor: I’d just look elsewhere hoping I could find someone competent at a better price. I’m cheap, and I won’t let myself get gouged, but I also don’t want to have questionable plumbing work done based on low price.

You negotiate with a contractor by first deciding your priorities, in terms of material and also as it relates to the actual work. What is most important to you? Price, speed, quality, easy process, flexibility, project length, design assistance, range of materials available and other aspects not listed are all topics of concern. You only see the price tag once but you will see the quality of work and and materials daily for years to come.
Selecting a contractor is much like picking a spouse. Flirt with several, then do background check the ones you like. Go out on a date with a few then have the "Are we ready for a serious relationship?" discussion with one or two candidates. Pick the. most suitable candidate and negotiate the terms of surrender. If your values and goals don't seem to match with anyone, go away and do some soul searching and reevaluate your criteria.
If the price comes in higher then expected, tell the contractor and ask him what he suggests. If you have a set number, ask what options you have to get the price to fit in your budget. Can you compromise on materials, choose an easier path or eliminate some desired feature? Could you do some DIY like painting or demo?