Can I turn a large laundry room into a combination laundry/3/4 bath?

Sure you can. It's easier because all the plumbing is already there. Since this is a major plumbing job, I would hire a professional plumber. If the room is large enough, I'd put a wall between the new bath and laundry area for privacy.

I have a half bath with washer and dryer crammed in to a 'much' smaller space, so yes, I think with some planning, you could get a 3/4 bath in there with enough counter space to fold laundry on...running the water and waste pipes is really the only trick part of the planning, which is 'much' less complicated if this is on the main floor, and the pipes are exposed in the basement.

So long as you have access to repair, I see no reason why not........

This laundry room is on the second floor but it already has water pipes and right behind it is a bathroom. If the toilet drain can go into the same drain as the soapy water then I am hoping they can use existing pipes. I don't know anything about plumbing except that in an RV the toilet (black water) has a totally separate holding tank. I would think in a house that it would all go into the same sewer.

Anyone have any ideas on what something like this will cost me to do? I already have the sink so I just need to buy the shower and toilet.

Purchasing the shower and toilet will be pennies in comparison to the labour. To ensure that the project is finished properly to code, you will need qualified tradespeople in plumbing, carpentry, electricity, drywalling, painting, flooring, countertops, cabintry, finishing, and tiling if you choose to have any in the new space.

There will be costs along the way from demolition, disposal, clean up, possibly permits and inspections, and unforseen materials needed to integrate the 'old' with the 'new'.

Yes, it can be done, but you are looking at $10's of thousands.