There are some exceptions but the procedure is usually : get a job offer from an employer that is willing to sponsor you, then talk to a lawyer... But a lawyer might be able to help you find ways to immigrate without a job offer, but that seems pretty unlikely to me.
The H-1B visa would normally be one of the 'classic' ways to be authorized to work in the US in your situation, the problem being that in recent years demand has far exceeded the available number (annual cap). For example this year 199,000 petitions were received during the first week, for only 85,000 H-1B available (including 20k for holders of advanced US degree). Therefore the USCIS now holds a lottery to determine which petitions will be reviewed first, and once 85k petitions are approved, you have to wait another year to apply...
The cost for an H-1B is nil for an H-1B as the employer has to pay for it and cannot ask you to reimburse those fees. You might have to pay in order to have dependants (wife, kids) added to your petition though.
The problem used to be to find an employer willing to sponsor you for the visa (the difficulty varies greatly depending on your industry), now the H-1B visa cap makes it more of a time issue unfortunately.
Other options you might look into depending on your how long you would like to work in the US: J-1 for an internship, E-2 for a company whose 'nationality' is the same as yours, L visas for a transfer to the same company in the US, F-1 visa for studies, and the O-1 visa which I will let you research by yourself and/or discuss with an attorney (the qualifications aren't as difficult at they may seem, trust me)
It's worth considering the L-1 visa route - work in the UK office of a US-based company for 1 year and you become eligible.
One disadvantage, you are bound to the L-1 employer in the US unless you can get a H1-B, work visa or family-based visa (yes, I found myself a green card wife of nearly 10 years now).
The L-1 process was a lot of paperwork, but the sponsoring company paid. The rest we did ourselves without a lawyer. Big fan of http://www.visajourney.com - lots of folk in similar situations and howto guides.
Happy to talk more if it's helpful, either here or my username at gmail.
The H1 or entering education in the USA are probably the two main methods, but even those have gotten significantly more difficult as others have pointed out (H1 cap etc).
Whilst not immediate, a lot of the other visas offer a faster or more direct path to entry in the US.
If you are graduating, not necessarily sure what you want to do and you see a potential US move a few years down the road then I think the L1 route is still a very good option. In my case I had worked for a big company for a few years and applied for a new role internally, visa process was painless and I think took around 6-8 weeks total - the L is not subject to caps so once your company is willing to sponsor, things tend to move rather quickly.
As a skilled software developer with a relevant UK university degree (3 years BSC) and work place experience, interested in working in the US - What is the ballpark range of costs and wait time involved in getting a visa to allow me to work for a US company.
What's the general procedure, - get offer from job, then -talk to immigration lawyer, or the other way around?
Thanks!