Traveling by Train

There is little if any security for boarding trains. Platform information is generally available about 30 minutes prior to scheduled departure so no point in being earlier than 30 mins prior to departure.

The only disadvantage of advance-discount tickets is that they lock you into a specific route, date, and time, and are no good if you miss a train or change your plan. Some larger stations in England have a ticket check before you enter the platform area, but that won't be the same everywhere you go. In general, your first station may be a "blind" experience but you'll get to know the others as you arrive, and will probably depart from the same station later. If leaving from a bigger city station in the UK, see station maps at http://www.nationalrail.co.uk/stations_destinations/default.aspx.

Your question a few months ago included travel in England, too, and a preference for flexibility. It's not clear to me whether those ideas have changed as the plan solidified.

See our general notes on European Train Stations and Getting on the Right Train.
In Ireland, you need not buy far a head to get a nice discount; just buy online at www.irishrail.ie rather than in the station.

"Are there security lines or processes we need before we can get on the trains?" - no, with the one notable exception of Eurostar trains. On some stations there are automatic ticket barriers you put your ticket in to gain access to the platform, but these are not universal and many stations do not have them.

"How much time should we allow before our train leaves to be at the train station?". Depends how big the station is, how long it takes to walk through it. So long as you are on the train before they lock the doors (up to 60 seconds before departure time, often less) then you are OK. For stations I am familiar with, assuming I have a ticket, 5-10 minutes. For stations I am not familiar with, 20 minutes+.
Maybe more if you have kids to herd or want to buy food or drinks to eat on the train.
Once you get used to them, stations are easier to navigate than airports.
1) Walk into the station, look for the departure board. Photo of the one from Glasgow central: https://upload.wikimedia.org/wikipedia/commons/0/00/Departures_board%2C_Glasgow_Central_railway_station_-_DSC06291.JPG
2) Find your train on the board and wait for the platform to be listed (10-15 minutes before departure). Zoom the photo to full size and you can read the board.
3) Go to your platform. You can see platforms 2, 3 and 4 under the sign.

Thank you all. You have been most Helpful. We are also traveling a little in England by train. We have decided to make reservations before we leave.

I hope the OP doesn't mind but I have a travel by train question- Do the trains (and buses) get extremely crowded in mid-to-late April? I want at least some sort of flexibility while traveling but note that there are disclaimers on the transit sites about tickets potentially selling out.

Small sample size, but my wife and I took the bus direct from Shannon airport to Galway, and back a couple of days later, on our trip in May 2016. We just bought a ticket right before the trip and the bus was fairly empty. Only thing about buses is the legroom is very poor, worse than a plane as I recall. Luckily I sat in an aisle seat and was able to stretch out. Also took the bus round trip from Cork to Kinsale in August when I was staying over at the end of a business trip a few years back. I was waiting for the bus 45 minutes prior to departure, then a few minutes before there were suddenly forty people in line, a place that I was not standing because I did not know where it was the customer to line up. The bus filled up and I barely got on. On the way back, I was worried about getting a seat but there was a reasonable amount of room. I don't think you can reserve a seat on buses. You buy a ticket and after that it's first come first served to actually get on the bus. Maybe it's different on different routes though.

Thank you! I'll make sure to show up a bit early for my intercity bus trips to ensure I have a spot on the bus!

Good to know. We will also be taking the bus from Cork to Kinsale and Galway to Killarney and Kyle of Lochalsh to Fort Williams onto Oban (Scotland) in September. We will each have a carry on and a small personal bag when we get on the bus. Are we going to be able to get on the buses without checking our bags. Is it going to be a struggle getting on those buses?