I too have always done all my own travel arrangements. Living overseas for 22 years makes that a very comfortable process and I believe I can do far better than any travel agent I've ever run into, especially now with the likes of the airline bookings systems and no commission fees etc.
In Europe we have tons of low price airlines that if you book tickets in advance allow you to get pretty much anywhere at ridiculously low prices. Of course baggage comes on top of that in 99% of the cases so one needs to carefully weigh the extra costs of bringing a golf bag against the convenience of traveling with your regular airline if that exists. In the case you do have a regular airline, i.e. I live in The Netherlands so KLM has the most direct flights and will always be one of the few competing airlines with the cheap airlines. Being a frequent traveler with them has certain advantages that can be worth slightly more than the difference in price all things being considered. For example, KLM has a virtual Golf Club - with a free membership. However, what many people don't know is that being a member allows you the opportunity to bring your clubs for free, anywhere. That can be a big difference. On top of that, lounge access and being able to go through priority security save time an hassle in terms of getting to the airport. In Europe I can normally arrive, check in and go through security in under 15 minutes. (This can be different in the US or London for example).
Regional/low cost airlines can save tremendous driving times when looking at multiple country trips. Easyjet, Transavia, Ryanair etc etc have strong offerings from major European hubs can flights can be as low as 25$ one way. Book it into a package with your round trip and you may pay literally hundreds for this through a travel agent who just books you on BA or United for example. One example, I once (4 years ago) flew from Dublin to Sligo (?) (close to where the Buda will take place) for 7.50 euro. Paid 25 euro per direction for my golf bag. Sure that may be a little more now but just an example. Or flew roundtrip Amsterdam London, Amsterdam Edinburgh, Amsterdam Manchester for under 100 euro.
In terms of accommodation, Booking.com is a great place to start as it also allows you to check proximity to your destinations. For obscure destinations check with the clubs or on their websites and often you find preferred B&B lodging or the like, unless you just want to opt for fancy hotels. Then it's a certain no brainer. Again it's all about budget.
Rental cars, a simple cross reference. Start with something like rentalcars.com then compare with any variation you like, Travelocity, Google and don't forget to check the direct websites of the rental car companies as this can save a lot or get you better cars.
Golf courses are rarely in cities so public transport makes things pretty complicated. I always prefer driving with golf related trips. There is far more flexibility and less reliance on fate and connections.
Advance planning, some people plan way way in advance, 6 months to a year. I prefer to travel in low season, thus not to the UK, Ireland, Europe, USA etc in July or August. I avoid kids vacations as a time to travel for golf. For me the best time to go to UK/Ireland is March/October. Sure it might be a little colder, you might miss out on some ball losing rough but that's a trade off I'm willing to take.
In terms of baggage, I don't see any inconvenience whatsoever traveling with my clubs. You only really have to worry about getting them from baggage to rental car. Make sure you can roll them. That's it. Unless you travel with carry-on only. But I'd rather not play than rent clubs.
Credit card, yes should cover insurance. That's tough to find in Europe/easy in the US I believe. American Express Platinum Card does in Europe, plus it also doubles your miles with KLM/Air France (this is just one example). I even rarely travel with any cash. Just make sure I have Amex and Mastercard. Withdrawing money from a cash machine will afford a better exchange rate in most cases than worrying about cash. Also an option if you are going to visit local friends that you send them some money in local currency to give you upon arrival. If so use Transferwise, they trade at the market rate and charge a small fee for the money transfers, saves a fortune from anything you can do on your own or through your bank, you can send pounds/euro/dollars to your local friends and pick it up upon arrival from them and end up picking up several hundred more than you would of had if transferred through a bank. Just a thought, and not for everyone but it's no hassle whatsoever.
Personally, I map out my golf first then plan the rest backwards piece by piece, then go back and fill in the blanks, ad courses etc based on time and access. Regularly for 15-20 rounds a trip at different courses so clearly for me this process works out. I write nice letters to the club managers if I don't have friends/acquaintances there. Ask advice when necessary and try to be flexible in my requests. Sometimes it works sometimes it doesn't. Sometimes you have to call to follow up after sending the letter twice. Your approach and discussion should be scripted unless you are naturally gifted with selling your own USP's of visiting ;-)
As a side note. I'm commenting more for Europe/UK/I/US and Australia etc. Some more complicated destination require another approach. Japan for example I'm heavily reliant on my friends there. Same for Korea or China. There it's much more relevant to win in information from people that really know their way around. That goes double for accessing the top courses there I guess.