The following is just how I personally work and see UX+UI+Graphic Design. Mine is just 1 of many viewpoints.

It's complicated:

1) Grouping all designers together and assuming we're all good at design, graphics, ui, ux, and info architecture is like grouping all programmers together and assuming they're all good at php, C++, java, ruby, mysql, oracle, javascript, visual basic, python, and perl. See what I mean. When you want a designer, you need to understand WHICH type of designer do you want. The same way you need to know what kind of programmer you need. In many elements of design such as Typography, layout, logos, branding, color, information architecture, visual graphics, user experience, and interface design. Finding a designer that does ALL of these well is extremely difficult. And it may not be a good idea to hire them either. Would you want to hire someone that has a little experience in every commonly used programming language or someone who specializes very well in just 2?

(All designers have a few necessary overlapping skills and often can work together very well and smoothly if they stay out of each other's way)

- UX Designer - In charge of what the user "FEELS" and what the user will get out of the site. Sets up user experience guidelines, helps create, pull, or finalize all features, user stories, usability studies, navigation maps, wire-frames, prototypes, persona storyboards. If the UX designers does their job wrong the user can "feel" confused about the site or doubt its legitimacy or ability to solve the user's problem and within 5 seconds leave, never to return. So emotion and feeling is extremely important. The UX designer crafts the foundation of the site based on predicting what the target audience wants and how they expect the site to help them achieve it.

- UI Designer - In charge of "HOW" the user interacts with the site. Sets up user interface guidelines, eye flows, layouts, button and input controls, control over all visual elements and what they do and how they interact with the user and a lot of overlapping responsibilities with the UX designer.

- Graphics Designer - In charge of what the user "SEES". Branding, logos, company colors, typography and custom fonts, sets up color & visual guidelines (buttons, logos, etc...), frequently works with photos and vector illustrations, often photographers.

See how different designers can be? In general you want to start with the UX guy, then the UI and Graphics Designer. You can have all 3 in the room but I personally believe the UX designer should ALWAYS be the first in.

2) Chasing the mythical designer + coder + know-it-all. Some of us can do design very well and a bit of front end coding. Design (visual) and programming (logical + mathematical) are so different. All the good programmers I've seen are terrible with visual design and all the great graphics designers I know are bad programmers. This is for a reason. It's very rare that someone has acquired both talents.

3) Don't make your designer do ALL of your front end coding. Because that just wastes everyone's time and assumes your designer is an efficient javascrip / jquery / css + xhtml coder. I personally love to start with css+xhtml THEN photoshop if needed. But as soon as I hit heavy javascript coding I start sinking. I would love to be able to get my design done and pass it on to someone who can code up the javascript quicker and better than I can. In a startup environment where speed and iteration are crucial this is really important. Everyone specializes in something and does that something very well and very quickly. Rather than having only a few people try to do everything they're not good at and end up doing a mediocre job that took three times as long as it should have.

4) Your broad requirements make us feel insecure. I'm a user interface + user experience + information architecture type of designer. Color, graphics, logos, custom fonts are NOT my strength and I cannot in any way compete with graphics artists who specialize in areas. I can do front end coding but only to an extent. I can make really easy to use sites but I can't give them that color + icons + logo + custom fonts + graphics themeing that a graphics artist can. Likewise the web is showered with websites made by graphics artists who aren't good at ux + ui + info architecture. These websites look pretty, have great icons, colors, & buttons but the layouts and site architecture are horrendous which leads to the user experience being horrendous as well. So when you ask for a designer that can "make everything pretty" I immediately skip the job offer because I don't want to disappoint you. When a visual designer sees the job opening they'll send you their portfolio of pretty websites that have improper eye flow, and terrible ux and you'll decline hiring them. Had you taken out that "graphics designer" must-have-requirement I would have gladly spent hours chatting with you in person (never online) on the direction you can take.

5) Hire me (UX + UI + Info Architecture) BEFORE you've hired a general graphics designer (icons + colors + logo + custom fonts). Otherwise It's useless. It's like hiring a painter before you've got your drywall put up.

Long Story Short, many startups don't know what kind of designer they need. If they hire a UX+UI designer they need to do that at the very beginning, then after they've got their beta product/service tried and tested they should go for a really nice graphics designer to help them with branding and look and feel. If you do it the other way around you just end up wasting time, and cause a lot of frustration for both designers because they have to RE-do all their work when ironing out design conflicts.

A UX designer is someone that you need to hire at the very beginning. You would sit down with them for Hours and hours over the course of MANY days (so pick someone you can stand) with huge sheets of paper and will start drafting out what it is the site MUST do and what it must NOT do. Finalize requirements and features, list content required, intents, user stories, and a million other things. There will be a lot of compromises having to be made, and there will be a lot of back and forth decision making. You will talk with your UX designer more than any of the others.

A UI designer is focused on the users, not you. They'll be off doing usability and predictability studies and designing the interface that the users will be using to solve the problems that the UX designer outlined.

A Graphics Designer can telecommute more, and can be left alone, send you mockups and seek feedback and a/b testing from users. They'll take everything from the UX and UI designer and add emotional appeal using color, lines, graphics, imagery, and photography, typography, logos, & icons.

All 3 are very important. If just ->1<- of these designers does their job wrong your website will be off (which may or may not hurt you). Look at craigslist, it's visual design is horrendous but it's ease of use thanks to a decent UI made up for it. Unforunately there's lots of spam and other problems that is killing' it's user experience.