HSG and IUI are natural first steps. I did both (6 IUIs) and was not successful, but we still don't know why.
When I had been trying for over 2 years, I got pregnant 8 weeks before starting IVF. We were saving money and waiting until January (to use our FSA funds) and we'd already attended a few classes. And in early November I got a positive test. I'm still not sure why exactly. That's the problem with unexplained. I was healthy, my DH's sperm looked fantastic, my uterus looked normal, but nothing worked. Until one random time it did.
We stopped using protection just before my DD turned a year (I weaned at 9 months, so my cycle was back). I got pregnant the same month she turned 1. I miscarried, unfortunately. Then I had a chemical 3 months later. And two months after that I got pregnant again and am now 17 weeks along. I have NO IDEA why I couldn't get pregnant for over 2 years and now it seems I can get pregnant every few months. I suspect I might have had endometriosis, that the pregnancy helped clear, but I'm not sure.
Try to IUIs, but just do 3. Then collect yourself and consider IVF is the right route. If my endo had been diagnosed, that would have been the path the REs told us to go anyhow (IVF) so we got there on our own without a diagnosis.
Good luck. I hope it works out for you soon.