Most often the main topic that a client wants to know is "what is this going to cost me ?" But in point in fact, that is one of the last questions to ask, these are some of the first.

Do they have the ability to perform any programming that may be necessary to complete your site?
This can run the gambit of java, to flash, to database programing, which of course depends on what you need the site to do, and what your willing in some cases, to pay for. More complex skills, often mean a more complex rate.

Discuss these things frankly with a perspective designer, as skills and preferences ( html, CSS, XML feeds etc. ) vary. The first few you connect up with may or may not be able to provide what you want, no harm done, just shop till you find one that can give you want you are after.

Will the designer include meta tags and other behind the page services ? Will there be extra fees for this ?
Realize, some firms will break down each part of your site, into its own payment category, whereas others, will quote you a flat rate for the entire affair. Others charge by the hour, some by project section, know what is included and at what rate.

Do they understand what your business does and who your customers are?
This is important, does the designer "specialize" , or can they do a general site. Do they understand your client base ?
Check their prior sites, which will give you a good indication of their area of expertise.

Can and will they work around a logo you already have ?
Are they able to design a logo for you if you do not have one? And if so, will there be an extra price ?
Further, will you "own" the rights to said logo or trademark ? The same goes for any other artwork created for the site.
Will they do the design using a webset made by others, or use their own ? If they use one they make, do you own it now ?

How long they have been in business ?
Given the length of the Internet as we know it, no firm is going to be able to say, they have been in business for the past 20 years
2 to 5 years is a respectable number.

Are there references that you can contact?
Talk to those who have worked with them, see how they handled things, before you make a commitment.

Will they do site maintaintance and if so for how much ?
They should be able to detail this in exacts for the asking ... if they can't or refuse .. run, do not walk away.

If you have a deadline, can they meet it ?

Is the designer easy to work with ... talk to?
Can they explain what they are doing in plain language ?
If the person speaks nothing but geek, you are going to have a problem .. can they and will they explain what they mean in simple easy to understand terms?

Can they provide a secure shopping cart ? Either by their own hand, or a third party provider ?

While having the design firm create the cart by hand does have advantages in flexibility, doing it via a third party remote is actually preferable, for the simple reason do" you" want to try and delve into the database if something goes haywire ? The carts crashed and the designer is off on a two week vacation.. can you deal with this ? Can your business ? Third party e commerce is reasonable in fees, and takes a lot of the worry out of all the cgi and database tricks, that almost require a programer to handle, that may not always be handy. The coding for the remotely hosted cgi goods, can be kept on hand and anyone who can copy/paste can handle repair or replacement, if needed, if they know where to put the codes.

How long will it take? ( a typical site takes about 30/60 days to create, the major factor being client provided data and feedback )
Will they be taking on other projects while working on yours? ( which also will effect the time factor, as a rule if you want the designer to be handling yours and yours alone, expect to pay more for the exclusivity )

Do they all look the same?
If they all look like they came out of a cookie cutter, then unless you LIKE the cookie, you might want to reconsider

Do the sites load quickly?
If the sites are visually lovely but slower than next winter, not a good sign, as they have traded looks for speed, and on the net speed matters.

Are they easy to navigate?
Do you have to figure out what to do, or is it readily apparent to anyone ?

Do you like their previous work?
Their priors, give a good indication of skill levels .. do the scan from the oldest listed site to the most recent, as skills improve over time. Never discount the fact that the site is made for another, they may have put in elements at client direction, that may not be your cup of tea, but are none the less, exactly what the client asked for. Look at the design elements, titles, meta tags, check the view source and look at the codes.. are they clean and easy to understand ?

Do the sites reflect their clients' business?
The site should reflect the mood of the product or information, and target the viewer.. if the site sells say, perfumes and soaps, one would expect it to appeal to women .. does it ?

Does the designer custom create each site?
This goes back to the idea of a cookie cutter.. their will be some similarities between any and all sites, as certain elements are expected to be within a given range of user expectations, but within those limits, did the designer match the content ?

Both these factors come into play, you have ideas you want to use and a good designer should listen to that. But at the same time, you hired them for their expertise, use that and listen to what they say as well. If they are telling you that such and such is not a good idea, they just might be right, even if you have your heart set on it.

Does the designer have the knowledge to create a “search engine friendly ” site ?
This requires an in-depth understanding of how search engines work.

Do they know what keywords are ?
Do they know the basic keywords in common use for your kind of site?
Can they give you suggestions on how to use those words in the text you write up for them to use ?
Will they edit the text you give them, to make the best impact of keywords ?
Can they say how their design choices are likely to impact the sites search engine position ?

A large tip off, Can you find the web designer's own website on the search engines for their main keywords and where does it place ?

Special note: Often you will find the designers own site to be a simple affair, not a showcase as one might expect. The reason is typically they have designed their own for speed, and searchablity by the engines as a main factor. With the splash as a secondary. Inversely if their own site is major splash and dash, but all the sites they have made are plain jane, you might want to ask a few more questions. And consider, if they are good, meaning they keep busy, they may not have much time to work on their own sites ( the case of the shoe makers child goes barefoot )

Will they get you ON the engines ?
Yet another aspect, you can have the best site in the world, but its useless unless it gets ON the engines. Part of the deal should be at least minimal search engine submission

Questions to ask :
How many search engines ( apox ) will the site be submitted to for the cost?
How are the submissions done?
How will you know that submissions have been done ?

Does the designer know and understand client usability ?

Which in plain english means, will the party who makes your website know Usability standards ?
Do they understand how to make a user friendly site that will be easy for the client to use ?
Will the site be viewable in all browser types ?
All known monitor sizes ?
Can they discuss design tradeoffs they will be using across browser types ?
"Meaning will the site goodies only work in IE ? but degrade well in Netscape or Opera ?"
Do they know what will look different in those other browsers and can explain what will happen to given design elements ?

What tasks do you, the client need to do ?
The designer can not do it all, in fact, should not. Factors such as, getting and paying for a host, a domain name and such like, needs to be handled by the client in most cases. The same holds true for any other paid services. Images and text needed for all pages, need to be provided, at least in baseline, to the designer. Lists of inventory, pricing and any special services, need to be laid out in detail. These things should be made clear to you at the onset.

Site ownership and transferability:
Believe it or not, some firms arrange matters to where they retain the ownership of the site and its domain to were you, as the site holder, cannot move it or sell it. This is a common enough practice, especially on the, "build your own" hosting services, to where the design firm holds all rights to the site.

I have personally seen this in action, to where the party who "owns" the site, regardless of hosting, literally cannot access it or make any changes to it what so ever, without going through their designer, as the designer has locked up the host and domain.

If you cannot pick up the site and move it and or turn it over to another designer, you don't own it.

A good hint tip :

How long have they been using the Internet over all ?
If they remember when the Internet was all use.net, pure text with no graphics, they have watched the internet be born and are a lot more likely to understand how it works, as they have watched it grow. If they got online last year, well .. they might know a lot, but are not as likely to know as much, over all, as an experienced veteran.