First tip and cheapest answer I have is: Gray Sisters Taxi. This way of transportation is a wild ride, so I suggest gripping onto something. The Gray Sisters are notorious for being quite reckless drivers. They share only one eyeball and you'll find yourself watching in disbelief as they fight over that instead of paying attention to the road. But for those with a faint heart and stomach, this is not recommended. To summon them, it's quite easy and cheap. Simply take one golden drachma and toss it onto the road. Shout the Greek phrase: "Stêthi, Ô hárma diabolês!" which means "Stop, Chariot of Damnation." A taxi that looks like it's made out of smoke should appear within seconds.
Next suggestion: air travel. This means airplanes, jets, and helicopters. Even mechanical dragons work - if you somehow manage to construct one, which is unlikely. During my time partaking in a Monster Quest, we had our airplane tickets prepaid. Should you not have plane tickets prior to your arrival at the airport, and it's basically a bust. They don't come at a cheap price tag, but can be bought if you have that money available. Remember to bring ID verification like a passport, which you can acquire by having permission from your parental figure or guardian.
The next options are a few of my favorites, since it doesn't put quite a dent in my bank account. That is calling an uber or taxi! We can't use technology since it is like a flare, but a kind civilian would be able to call an uber for you. If not, getting a taxi around larger cities isn't difficult at times. The only downside is that these options tend to only get you so far before having to turn around and go back. You certainly can't uber yourself across the country. Trains do go to more remote places, and they are quite cheap as well.
If none of the options above work out for you, I still have one left. But take this one with caution, and use it only in times of need and extreme desperation. Or if you need to get away fast. That is to steal a car. I don't really recommend this one; it's rather tricky. For older cars, it's easier to pop the lock and hotwire. Newer cars, on the other hand, require a lot more effort. Some tips are to look for a spare, hidden key or jump into one already running. It'd be handy if you had a Hermes kid able to pick locks, or a Hephaestus kid able to make a scrap getaway-mobile. (Do not attempt IRL. I do not condone stealing cars or committing any type of crime.)
With that, we have reached the end of the blog. Remember to stay safe and keep these questions in your head while you're heading out for an adventure. See you all on number 2 of this post series!