Although I'm going to pay particular attention to Iceland, this advice is still universal with reference to solo travel as it is this group of new explorers that I initially created the Iceland travel blog posts for.
I have found that learning how to adapt to changing circumstances has been a key development for me and given me growing confidence in travelling solo. As well as being open to the guidance of the Universe and trusting that I'll end up being where I need to be.
Flying
My first recommendation is to only use the Icelandair website if you are going to fly. It doesn't matter how much cheaper third parties are, if there is a problem then it's likely going to be harder to solve that problem third party. I have tried a few options and by far the easiest and stress-free journey I had was when I booked directly through Icelandair. Prices of flights do fluctuate a lot, and sometimes daily. I kept a spreadsheet once of how much those prices changed and it varied between £80 to over £500 for the same flight. I signed up to a 'flight alert' dude who sent emails about when flights were at their lowest prices for certain destinations. And I actually ended up paying more following his advice of when to book. I suggest tracking them yourself and just watching the prices go up and down, to see what kind of range to expect. Google flights do alerts to tell you when the price changes, which is well worth signing up to.
I always suggest adding hold luggage, particularly for Iceland. Two people can share one hold bag if you want to save a little money, but you will almost always need more than your cabin bag. And keep your eye open for any reward systems that are active. Even if you don't think you'll use them.
If you are an eco-warrior, consider offsetting your carbon emissions. There are some companies that plant trees. I cannot recommend any at the moment of writing this, but urge you to do some research.

Getting to the airport
There are so many options available to us now, and sometimes very competitively priced. One flight I had booked was leaving from Manchester airport. Every transfer option I checked was around the same price;
Taxi on the day
Train the night before and a hotel
Train on the day
Coach on the day
In the end I chose train and hotel. I got to have a leisurely breakfast on the day of my flight and arrive at the airport within ten minutes of leaving my hotel room. Well-rested, and carrying an extra sandwich I had made and packed from the breakfast buffet.
I suggest using Trainline to find your perfect journey. Also compare your journey to the cost of a Railcard. Sometimes they can pay for themselves in one trip and then you get to use it the rest of the year.
Coaches can usually offer a journey at a much cheaper price than a train. Hello London for £5! The journey is longer but the saving can be worth the extra few hours. The other perks of coaches are that you can recharge more comfortably and easily on them than a train. There's less noise, less seat switching. My trick is to put some earbuds in, sometimes without music playing and just stare out of the window, dozing in and out. I recommend National Express.
Ferry
There are other ways to get to Iceland. I see you can get a ferry there now via Demark. I have not tried this but will update you if I do.
Once in Iceland I always recommend the Flybus when travelling to and from the airport in Iceland. The system can be a little confusing for your first time, so I will walk you through it. You can book your tickets online before your trip for ease if you wish to. Upon arriving at the airport you don't need to rush to the Flybus. There is one always outside waiting and they only leave the airport once full. There are usually more parked behind them, waiting to collect the next hoard of passengers. So take your time in duty-free. Collect alcohol and chocolate if you need any. Just remember that the full journey between airport and hotel may take some time, so balance it up with shopping.
When you decide to embark the Flybus, they will usually take you to their bus station first, called the BSi. It can look a little extra confusing at night amongst lots of Icelandic yelling, but don't worry. They will make sure to get you to your hotel even if at first you feel a little like you just got home from deployment at the army. There are mini buses and coaches split amongst coloured zones. Depending on where your hotel is you will have been assigned a colour and will need to find that coach or mini bus, or will be instructed to stay on the coach that collected you from the hotel and continue to be driven that way.
I cannot stress this enough about the Flybus, you must relax and listen for instructions. The drivers get questioned constantly by travellers wanting to know how long it will take to arrive at their hotel and eventually they stop answering anyone. Even if it seems like it's taking all night, and seems like everyone is reaching their destination but you, you must relax and just let the journey unfold in its own time.

Travelling in comfort
I'm finally going to touch upon how to make the journey comfortable. Travelling solo can be scary. I cried four times on the way to my first trip to Iceland. I was terrified. Absolutely nothing went wrong but I was just scared of being alone. These things helped me and I use them whether I travel solo or with others;
iPod with a playlist of confidence boosting songs and a decent pair of earbuds.
Self-soothing techniques i.e. reminding myself that all is going to plan, everything is running on time, I have time etc.
Planning plenty of time for each stage. So when I went to cry in the toilets at the airport, I knew it wouldn't make me potentially miss my flight.
Choosing introvert friendly travel options. Such as booking the ideal seat on the train (window, near a luggage rack and toilet is my preference).
Researching everything thoroughly beforehand, such as watching You Tube videos, reading advice on Tripadvisor.
Staying in touch on the way with supportive friends and family by text. At times it felt like they were with me as we would laugh over something silly that I had seen.
Celebrating each stage of the journey. I do mean every stage. Getting on the train in your home area - that is a win!
Reminding myself why I booked the trip in the first place. What is your why? Keep hold of that.
Good luck! And as always you can seek my help.
*Some links may be affiliates and generate money or gifts. I only share links to services of products I use myself and will update links if I find better – Clair