How to easily switch to driving on the other side of the road?

When I learned to drive in the UK after years of driving in Europe I found four main issues:

1. The traffic rules (junctions and roundabouts) - this was actually the easiest thing to learn. If there is traffic on the road, you just follow other cars. Otherwise you need to remember it is basically same as in Europe, just opposite.
2. The position of the car in the lane - the trick here is to always align the car to the edge with the steering wheel, so look at the left side of the lane in Europe and right in the UK. Once you know the trick, it is easy.
3. Mirror positions - this is something that took me longest to adapt to. Initially I was feeling like during my first drive without the driving instructor in the car - I just had no idea what happens behind and to the sides of the car. Here I can just advise practise.
4. Gear stick - its on the left, as opposed to the right. It takes time to learn it, I don't think it is a major issue though.

I don't think you need lessons, just start driving more carefully than normally.

Something to remind you:

Seen in a rental car in Ireland, a sticker on the wind screen reminding the driver to drive on the left.

If you do rent a car without such a sticker, or take a car to where the driving is on the other side of the road, you can make your own 'sticker'. My brother used a plaster (like you put on your finger when you cut yourself) in the spot where you would look when driving on the right but not when driving on the left.

It has to be just out of your normal field of view when driving where you are suposed to drive, but anoyingly in view when you drive on the wrong side.

Tips for getting started:

Most people prefer driving on motorways to get used to driving on the other side of the road, as most cars go the same direction, you 'only' have to get used to cars merging from the other side and exits (off ramps) on the 'wrong' side and you might select to go to the slowest lane before your exit comes up, or just stay in the slow lane while adjusting.

Have a box talk to you (GPS-Satnav-TomTom-Garmin), telling where to go, so you do not need to worry about keeping track of the route. Set the machine to the system you are used to (American English if you are from America) but to the distance system of the area, (so mile where miles are used but km when in a km country.)
That way you will get the words you are used to but the distances which are also on the signs, and likely on the dashboard of a rental car. If you drive your own you should consider what is easiest for you.

If you have a passenger, get him or her to remind you to keep left when you drive away from being parked. Yes, it will get anoying, but driving away from parking is one of the points where it is easy to go wrong, specially early in the day and when you do get tired.
If you travel alone, you might program your phone to give an alarm with a 'keep left' as ringtone for the alarm, at the time you expect to start your driving. (Make sure it does not need action, in case it goes off while you have already started.)

Other moments you need to be very careful:

When turning one direction a few times in a row, you are more likely to end up in the familiar location on the road. So tell yourself 'keep left' every time you turn a corner. This is even more likely when you are lost and looking where you need to go. (Satnav-GPS helps avoiding this.)

When driving on small roads, where you will usually be in the middle of the road when there are no other cars, you run a big risk of diving to the wrong side of the road when someone else shows up. You will have to adjust to the road but it might be better to be a bit off the ideal driving line just to keep you reminding yourself to stay on the proper side of the road. Go slower than you would normally do on a road like that, and that is also slower than the locals do.

More tips:

Keep yourself as aware of your surrounding as you can.
As mentioned in one of the other answers, do not drink alcohol before driving, even when the local laws allow you a glass or two.
If you like to hit the alcohol bottle/can when the driving is done for the day you might even feel better when not drinking much in the evenings after driving, so you do not have a sore head in the morning.
But also make sure you do take more rest than you would usually do, (even more than truck drivers need to take by law) so you stay from getting tired.

Keep hydrated. Have water at hand while driving and drink enough through the day, especially drinks without caffeine.

Have your drinks and snacks handy (or have your passenger hand them to you) and take proper breaks for anything bigger, for hot coffee, and all other things you need to do.

Rest stops might not be where you expect them to be, like in Europe they are often on the side of the motorway with their own entrance and exit, in the UK they are often just of a normal road exit, on a roundabout. So you need to be more aware what to look for when you need a rest stop. They might be spaced different from what you are used to, in low traffic countries there might not be petrol points with restaurants for a few hours, in heavy traffic areas they might be every half an hour or more often. The old saw of 'use it when you see it, it might not be there when you need it' might be the rule to go by.

In short:

If you keep aware what you are doing and go with the flow, it should not be hard to learn.

I do not drive, but have ridden a recumbent trike in the UK (from the Netherlands) and I have been a passenger/navigator with many friends and relatives going across to the UK or from the UK where you need to drive on the right.

After 20 years driving on the right, I'm now in Ireland driving on the left. It's not so hard.

Some of the more tricky issues:

1. Keeping your lane. With a wheel on the left, like I'm used to, the driver should be in the left of the lane. But with the wheel on the right you should be in the right. I find my self sometimes drifting left and I need to fight it.

2. When joining a large road from a smaller road (of course you need to give right of way), traffic on the near lane comes from the right, not the left. Several times I found myself surprised - where did that truck come from? Pay attention to these situations.

3. You start driving on a small road, which is all yours. You may forget yourself and drive on the right, or in the middle. Now suddenly a car comes - you might take the right instead of left. This normally happens at low speeds, so it's easy to correct.