Yes, a robust HMO functions a lot like the NHS. You go see your [in-network] GP for a nominal co-pay. If you need specialty treatment they refer you to [in-network] specialists for for further tests. Similarly to the NHS, provided the GP and specialists say it was necessary, the treatment should be covered. (barring things like a specific exclusion in your plan, which you should be able to read before you sign anything)
With HMO's the big complaints come up with out-of-network providers. For instance, if you are hit by a car and rushed to the ER (unconcious so you can't object) and the ambulance and hospital are out of network, the HMO may refuse to cover those charges. However, most HMO's recognize what an insane PR move that is and will advertise that they cover all emergency care.
As with the NHS, you simply won't get a breast augmentation covered by an HMO (or PPO afaik). My GP used to send in every mole he removed for biopsy as that would "prove" to the HMO that it was necessary to remove it rather than cosmetic (even if it was probably mostly a cosmetic/convenience request).
My co-pay for prescriptions was $20 and as of last December it hadn't changed. I don't think the rate is different for a child's prescription, but that would be laid out in the plan. I know people who don't bother claiming simple prescriptions because the cost of the medicine is less than the co-pay (doesn't make sense to pay a $20 co-pay for a $10 medicine iyswim).
If you have an HMO you likely won't be arguing directly with the insurance company about necessary treatment or visits. Like the NHS, it more or less works for a basic level of care and the things you hear about on the news are usually something specialty - not that you won't have any unusual circumstances, but that you need to be aware if/when your choice of company could be a problem. DH is more likely to need an organ transplant if his one remaining kidney gives out, so we won't sign up with companies who exclude organ transplants from coverage etc. I have a Danger Girl toddler, so I make sure that we live near an in-network hospital and the local ambulance company is covered. Etc etc etc.
If you're worried about it you should definitely check the Kaiser website (Kaiser Permanente is meant to be a fairly good one iirc) and see what levels of coverage they offer. If you or your kids have a chronic condition you will want to be careful of lifetime caps on treatment.
Also check the new healthcare legislation as that will change things quite a bit in four years.
Finally, a quick note, Kaiser Permanente is a REALLY big ins company, covering much of the western US, so the will have a large number of complaints online. Sometimes it's really useful to read the horror stories as they can give you an idea of how the system works and what might be expected of you.