Method to travel from The Republic of Ireland into Northern Ireland

It's very easy to take the train or express bus between Dublin and Belfast.

As far as car rentals go, you'd have to double check the policy to see if you can drive it into another country and check what the drop off fee is for returning it to another country, if you plan to do so.

Yes it is easy to take the train. We took it from Dublin to Belfast, stayed two nights and took the train back to Dublin.

It depends when you are going. Until early Autumn this year there is no through train due to track work at Belfasf.

But when the service does restart Irish Rail have requested permission to increase service frequency to hourly.

The trains the fastest at 2:15 However, train service has been quite limited since extensive work adding new train stations and now the miles of train tracks being built for the final approach to theses new stations is getting done.
It all adds up to switching to a manageable 2.5 hour bus ride.

Car rental agencies do allow drivers to cross the border, but they often will levy surcharges inthe hundreds of euros and pounds, on drivers who drive the rental car in both the Republic of Ireland and Northern Ireland.

However, train service has been quite limited since extensive work adding new train stations

One new train station has been added- Belfast Central- which is actually a replacement for the former cramped Great Victoria Street. When trains restart they will terminate at Belfast Central, which is adjacent to the Europa Bus Centre, rather than Lanyon Place as they have been until now.

So there will be seamless train/bus connections as well.

The closure isn't really due to the building work at Belfast Central either, but due to a lack of capacity at Lanyon Place during the work, as all local trains which previously terminated and started are currently having to use Lanyon Place.

If you'll already have the car then simply driving across the border to visit sights in NI is really no big deal, but you'll need to read the fine print on your rental car contract to ensure that you're compliant with company policies and that you've paid whatever (minor) fee they may charge for a cross-border trip ... which people do all the time without too much drama. You don't want to inadvertently void your insurance by getting this detail wrong.
This all supposes that the sights you want to see actually require a car - still the best way to get around in the more rural areas but usually an impediment if you plan to stick to the cities. It also supposes that you'll return the car in the Republic - doing a one-way rental and dropping it in the North can be very expensive.

In 1997, I rented a car in the Galway and crossed into Northern Ireland with it and returned to the Republic.

It's possible car rental agencies have depots both south and north of the borders. If you rent a car on one side of the border, you might be able to deposit it on the other side without having to drive it all the way back. Inquire at rental agencies.

Safety: Make sure you're ready to drive on the other side of the road: 1. The driver is always near the center of the road. 2. Left-hand turns are easy and right-hand turns are challenging. 3. Drive the speed limit or lower. 4. Look for the directional sign ahead of roundabouts.

Common sense: Don't speed near the border. On my return trip near Enniskillen, NI, I was probably exceeding the speed limit in a rural stretch when I saw a British solider on patrol. I slowed down, but shouldn't have been speeding in the first place.