NedRN

Small agencies and strike companies take the shortest amount of time (could be just hours). There are a number of ways to speed the process besides agency choice. Written references (I recommend at least three) are far far better than providing names and phone numbers, are easily verified (some agencies may not even bother) and allow easy shift to new agencies. I also recommend getting your own references on each assignment, early (to protect yourself) and several (to choose the ones that best reflect your performance). Managing your own professional portfolio is much better than depending on any agency, and while agencies also collect references/evaluations on each assignment, they will never share them with you. Having them in hand makes switching agencies much easier, thus no agency will share them. Getting references or/and verifying them is perhaps the most common slow down for agency onboarding processes. A form to hand to a referee is also much faster and easier than getting a formal letter of reference. Samples can be found for download on PanTravelers - sign up for a free membership and go to Downloads. These forms leave space for an agency to insert their letterhead.
Getting your own physical clearance annually with up to date vaccination records also cut down on time required.
Different agencies have different standards for background checks and how often they have to be performed (usually a new agency will require a new background check) but if you can obtain a copy of a background check in the last year, this can also speed things up (depending on the agency). Background checks run a gamut of just seconds to run online, to several days, and this varies depending on the agency. Again, smaller agencies generally take the faster road.
Keeping a record of your work history to submit to agencies reduces your own work in filling out paperwork - no need for resume or CV like detail. Now some (larger) agencies want everything done on their own letterhead - it looks better to facilities receiving your profile, and perhaps to the JC, but many agencies can transfer your submitted work history to their own because you are that valuable to them to do stupid stuff on their own dime.
You can find out on your own which agencies are fastest which is not going to help you initially, but may help you find agencies are best at filling fast moving desirable assignments. Putting all your eggs in one agency basket means much more downtime if an assignment cancels. I'd talk to a dozen agencies (yes, a lot of work) and pick 3 to 5 agencies that you communicate best with and have the least BS and go through the full signup. I'd suggest a mix of small, medium, and large agencies for best possibilities of choice, availability, and compensation. Then you have a plan A, B, and C. Keep them all updated with every assignment and be up front about your plan. Ditch recruiters that are not happy with your free agent status.
Admittedly, signing up with lots of agencies is a pain, even if you keep your professional profile up to date. So many if not most travelers find sticking with one agency is less work and are willing to take the risks and likely lower pay to avoid the hassles of new to you agency signup. Your choice. While it was in the golden older days of travel, I once had an assignment cancel in the middle, and started two days later at a new hospital in the same city with a new agency and stayed in my housing losing almost nothing. A bit lucky perhaps, as I had worked for only one agency for 4 years (so did not follow advice above except I did have all my references for every job ever worked), but the due diligence required was much less back then.

Typical OR internships are 9 months or longer at teaching hospitals. There an expectation or explicit contract for you to work 2 years.
Personally I stayed 3 years at the hospital that trained me but I wanted the extra year to gain more experience in open hearts. Then I tested whether my skills would translate to other hospitals by doing some agency per diem in my metro area before I took the plunge to travel. I can tell you that after umpteen years of travel, I have been super happy that I got all that training. It really served me well.
ER is more entry level than OR. You can get by with less training but more training always gives you a better practice basis. If you go for this route, it is better to start at a level one trauma center. Harder to do a community hospital (for OR too) and go to a teaching or trauma hospital - it limits your options for travel and your value to a manager.
Now for both ER and OR, you can go for less training, and practice at less demanding hospitals who don't need higher level experience and travel successfully. Less assignment options, and less money, and a higher possibility of unsafe practice environment due to lack of experience. Don't let any of that scare you if this is the way you want to go, you can turn down any assignment you believe you cannot handle safely (in fact your license requires it).
OR pays better than ER - pay as a traveler reflects your level of training more directly than staff. Free market economics. Conversely, there are more ER assignments than OR, so you will have a wider choice of locations needing travelers.
My best advice to you is ignore all of the above, shadow a nurse in both specialties for a couple of days, and choose the specialty you like best. Then deal with however you can manage to get into that choice.

Contacts? Perhaps you mean contracts. Every single hospital will tell you that they have enough vendors to meet their needs. Other than being an already large established agency with experienced sales people, the best way in is to do a side run. Your contracts are not a selling point in any case, it is your profiles. In your case, your wife has a hot specialty, one that is in high needs. I'd recommend that she calls L&D managers directly and telling them she is interested in working there. That should turn into a mini-interview (managers like to save time), and prime the pump by telling her you have a favorite agency and do you care what agency you use? The manager could not care less, she just wants staff. Now when you contact HR/staffing, you can tell them that manager so and so wants to use one of your staff. Send them her profile and go from there with a rate and contract.
By the way, the easier path is not to cold call, but to call a previous manager who likes you. That smooths the conversation a lot. By the way, your wife's agency may have a do not compete clause in her contract prohibiting her from working at their client facilities for some length of time, usually 6 or 12 months. If they find out that she is working for their clients, they may have a legal case to enjoin her to stop.
This industry has a very low barrier to entry, which is why there are so many agencies. I've never done local agency, just travel, but fundamentally the two are very similar. With local though, I think you need to have nurses signed up ready to work for you (and perhaps willing to go through the same process as above) to get some critical mass. With travel, one traveler is enough. There may be local or state based regulation, especially if you will be doing home health. Except for maybe two states, there is not much regulation for travel nurse agencies.
If you Google my screen name and "independent nurses FAQ 108" you will find my description of starting a travel nurse agency. While it is a bit dated, this resource is still pertinent and better than the $100 to $500 "start your own agency" scams. It should give you a good framework for going forward.
Good luck!