It seems to depend on what passport you have.
Both the tourist information website and the Ecran Airport site say the same thing.
For tourist trips of up to 90 days, citiziens of European Union (EU) nations, Turkey, USA, Canada, Mexico, Israel, Japan, Singapore, Switzerland, Iceland, Norway, Australia, New Zealand do not require visa.
It looks like all other countries do require visas, you'll need to contact your nearest North Cyprus Embassy to get one.
And if there isn't one, then it looks like Northern Cypriate matters will likely be handled by the local Turkish Embassy, so speak to them.
As for issues with the two Cypruses, if you see the "Crossing the Green Line" section on the tourist information site it suggests you shouldn't have any issues as the land border is open.
If you've not got a suitable EU passport or visa, you'll need to get a visa to cross to the south, and you can certainly ask about it when applying for that visa.
I wouldn't expect any issues though.
travel.stackexchange.com/users/26/Gagravarr
Just in case others are still looking at this question and answers - if you have any of the passports listed in answer 1 above, there is absolutely no problem at all flying into Ercan airport and then crossing to the ROC and back.
I have an Australian passport
and I have had a holiday home in TRNC for 8 years.
I work in the Middle East, I regularly travel to Europe, and the USA and Australia once or twice a year.
I also have a TRNC student visa stamp in my passport
and I have never once had a problem with entering any other country because of it.
travel.stackexchange.com/users/64962/McLean
The only legal way to fly into Cyprus is through Paphos (PFO) or Larnaca (LCA) international airports.
If you want to visit KKTC (Turkish occupied part of Cyprus)
you can use the crossovers in Nicosia, but don't fly into the Tymbou so called "Ercan" airport.
It's an illegal port of entry and EU Immigration
will likely hassle you when you return.
It also be noted that Turkey refused to establish direct communication with Nicosia ACC which causes serious concerns over safety and flight efficiency in the region.
Republic of Cyprus (RoC) will not let you cross over into the south if you entered the island through an illegal entry point such as Ercan Airport, in fact I know they won't.
It's best to abide by EU and international law if you want to avoid hassles.
travel.stackexchange.com/users/49997/Paul Morphy
If someone would like to visit Turkish Republic of Northern Cyprus then you must fly to Turkey in the first of all
and then you must take your transfer flight to Northern Cyprus,Ercan Airport.
At the border of Turkish Republic of N.C,they will stamp your passport.
If you've EU passport or USA or any visa-free country for EU then there will be no problem when you want to cross the border to Southern part of Cyprus.
Because Southern government has no right to act against EU laws so Southern Cyprus border police must get you in their land.
As long as you have EU passport or visa-free passport for EU then you can enter.
It's all the same for opposite way.
If you'll fly to Southern part of Cyprus then again,It's all fine.
Get yourself to the border and enter to the Northern part of Cyprus.
Only Turkish passport holders can not visit Southern part without a valid South part of Cyprus (tourist C type) visa.
And some other 3-4 more countries too.
If you have a Turkish passport then you'll need to get visa and then fly to Greece and then to Cyprus.
Cypriot passport holders could use opposite way to visit Turkey too and they can get visa on their arrival to the airport because of the laws between Turkey and EU.
travel.stackexchange.com/users/92288/Ottman Turkoglu
First I need to mention that I used to live in Cyprus.
My suggestion is to fly to Cyrpus on Larnaca or Paphos airports.
Visa is not required if you hold an American passport and plan to stay for less than 90 days.
Crossing into Northern Cyprus is very easy.
I used to cross from Nicosia (Leedra Street) on foot, however this is not the only crossing point.
Be aware that if you are flying to Northern Cyprus Directly, (via Ercan airport) and want to cross into Cyprus (South), you will be considered by the Government of the Republic of Cyprus to have entered Cyprus through an illegal port of entry.
The Government of the Republic of Cyprus reserves the right to fine you for illegal entry if you cross into the south, or decline you entry into or exit from the Republic.
Also, another point: if you are planning to rent a car in Cyprus and want to cross into the Northern side with it, it might not be allowed due to insurance issues .
So ask about this before getting the car.
Cheers and good luck!
travel.stackexchange.com/users/92271/Nick
I know that Northern Cyprus is recognized only by Turkey, and not other countries. So how can I visit this republic - only from Turkey, or there are some other ways? Will I have problems with getting Cyprus visa after such trip?
Northern Cyprus - way to visit and what problems can I have with Republic of Cyprus after that?
travel.stackexchange.com
<visas><customs-and-immigration><cyprus><disputed-territories><northern-cyprus>
travel.stackexchange.com/questions/790
travel.stackexchange.com/users/19/VMAtm
For trips of up to 90 days, toruists from the EU, USA and other visa-free countries will be able to cross from the north to the south of Cyprus. However, you can only fly to the north from Turkey and to the south from Greece, and Turkish passport holders will not be able to cross via the land border. Only Turkish passport holders can not visit soouthern Cyprus without a valid visa whilst citizens of Europe, America, Canada,  Mexico, Israel, Japan, Singapore, Switzerland, Iceland, Norway, Australia, New Zealand do not require a visa for tourist trips of up to 90 days..However it is important to remember that firstly, the only legal way to fly to cyprus is through Paphos or Larnaca international airports. Secondly, if you want to visit the Turkish republic of Northen Cyprus you have to go through Turkey . Lastly, it is considered illegal to cross over the border to southern Cyprus after flying directly to northen cyprus.. 
Citizens of Europe, America, Canada,  Mexico, Israel, Japan, Singapore, Switzerland, Iceland, Norway, Australia, New Zealand do not require a visa for tourist trips of up to 90 days.  Firstly, the only legal way to fly to cyprus is through Paphos or Larnaca international airports. Secondly, if you want to visit the Turkish republic of Northen Cyprus you have to go through Turkey . Lastly, it is considered illegal to cross over the border to southern Cyprus after flying directly to northen cyprus..  Only Turkish passport holders cannot visit southern cyprus without a Valid Visa.