First of all - the decision to begin praying the LOTH has been one of the most profound and life-changing decisions that I’ve ever made. I would strongly recommend it to anyone - in fact, I would particularly recommend it to laity, who don’t have as much exposure to their faith on a regular basis as the consecrated religious have. The trick, is not to burn out, which is the big issue that most beginners to the LOTH experience. Start slow, and take it easy. Remember that there is absolutely no obligation at all to recite the LOTH, so don’t beat yourself up if you miss an Office, or didn’t quite recite one correctly.
As for where to start, officially, the Church considers the most important hours to be Lauds (Morning Prayer) and Vespers (Evening Prayer). The Church considers these to be the primary “hinges” of the liturgical day, on which all the other hours rest. However, I’ve found that most people advise newcomers to the LOTH to start out with Compline (bedtime prayer) - mainly because it’s a very short office that almost everyone can fit into their schedule, without too much pressure. Try making a promise to yourself that you’ll pray Compline every day before bed. Then, once you’ve gotten the hang of things, try to fit in Vespers after the dinner clean-up. Just remember, the important thing is to avoid burn out - which means not pressuring yourself too much.
I, personally, would not advise you to get a print Breviary before you get the hang of things.
Divineoffice.org is a great website that contains all the offices in a very accessible format.
Universalis.org also is a good recourse. IIRC, the primary difference between the two is that
divineoffice.org is based on the American LOTH, while
Universalis.org is based on the British Divine Office (although I’m not sure if this is true or not, so please correct me if I’m wrong). Personally, I like the American version better. If you have a smartphone, there’s a great app called iBreviary (this also uses the American version). Universalis and
Divineoffice.org also have apps, but they cost money, and there isn’t much point in paying for them when iBreviary is free.
Once you get the hang of things, I would greatly recommend getting a print Breviary - either the four volume American LOTH, or the three volume British Divine Office. It’s very technical stuff however, so even most people who think they have the offices down pat get a bit confused at first. And once you get into the rhythm, there’s always room for expansion. Once you have more time on your hands, you can look into using a one-week psalter rather than a four-week one. This essentially means praying the traditional 1962 Breviary, or a monastic one.
Divinumofficium.com has the 1962 Breviary online, and there’s an app for it that costs 99 cents (Breviarium Meum). Baronius Press publishes a print version, but it’s quite an investment at $350. If you can’t quite afford that, and are willing to go a bit more exotic, you can also look into an Orthodox Horologion, which uses a one week psalter as well (although the offices are ridiculously long, so make sure you have time for it). The (majority of the) offices can be found at orthodox.seasidehosting.st/seaside/home, and I won’t get into print Horologions here, but they’re available as well. I must add as a disclaimer - the Horologion is used by Eastern Rite Catholics as well as by the Orthodox. So, well it’s not generally accepted in the Western Rite there is, technically, nothing “wrong” with it, and I recommend everyone try it out at least once, because it really is quite beautiful.
Hopefully I’ve set you up with enough info here to get you started.