It used to be possible to cross to the Ethiopian side of Moyale without a visa for a few hours to see what you were missing (which isn't really very much at all), but this is no longer permitted. The border is open from 6am to 6pm daily. If you are entering with a vehicle, you'll have to visit Ethiopian Customs. The process is all a bit confusing, but the officers will want to see that you have car insurance that covers the amount of the car or, failing that, will ask you to pay a security deposit. They were unsure how you would get this deposit back if you are not returning to the border, but were confident that, somehow, it would work itself out…
If entering Ethiopia, be aware that visas aren't available at the border, nor at the Ethiopian embassy in Nairobi for non-residents of Kenya. Also be aware that you cannot use an e-visa to enter Ethiopia via borders – e-visas are only for airport arrivals. The Kenyan Border Post was being moved to a new building at the time of research, but you should still expect a bit of a wait before you're allowed to leave (or enter) the country. The Ethiopian Immigration Post is tucked off the road as you enter Ethiopia – ask locals for directions and don't even think about entering without a valid visa.
If you’re coming from the other way, three-month Kenyan tourist visas (US$50) are available at the border for most Western nationalities.