Best way to travel the Appian Way from Trastevere?

Probably the simplest, quickest, and easiest way is to just grab a taxi to the entrance of the Parco Regionale. You could do it via the metro connecting to a bus but that's probably more hassle than it's worth for the minimal savings involved.
This place is located at the entrance and gets good reviews. You can either rent bikes independently or sign up for a guided tour ... which might be the better option to get the most out of the experience.
If going indepenently you can pick up a map from the bike shop that'll get you on your way.
Used to be that the favored day to do this was Sunday when vehicular traffic is restricted. Not sure if that's still the case but it'd be worth confirming before you set off.

I'll agree with Robert that taking a taxi to a jumping-on point would be the most efficient method. As far as which point to choose, and to walk or bike is a tougher question. We started at Casal Rotundo (#49 on the pdf map I've linked below) and were happy to have done it on foot versus wheels but that was personal preference. I will say that the further out you go, the more you feel like you're in the countryside versus a city.

The website lists some specific places that can be visited along with way that have closure days and/or fees:
https://www.parcoappiaantica.it/smart-info-ita-eng/

This is the map we used to tell us about the other interesting ruins we encountered on our walk. I believe this is the same map that can be obtained at the bike rental office mentioned above but as we did the walk backwards - walking back into central Rome from our jump-on point further afield - instead of the other way around, we downloaded and printed it out in advance. I don't know as I'd use it for the listed hours for some of the attractions though; use the link above for that as it's likely the most current.

https://www.parcoappiaantica.it/wp-content/uploads/2016/05/AppiaAntica_eng_2015.pdf

Here's one frequent RS poster who describes their 'adventure' on 2 wheels instead of their two feet, and some others weighing in on that method.
https://community.ricksteves.com/travel-forum/italy/biking-on-the-appian-way-be-forewarned

I purposely didn't describe how we got to our jump-on point as I wouldn't recommend that way to anyone else - I'd take a cab - but if you're OK with a long day, you could even take an airport bus to Ciampino and walk in from there; the road runs close to the airport, and there's a jump-on point that would likely work at Via Dell' Aeroscalo. That's probably what we'll do next time as it's not that much further out than Casal Rotundo.

We made a long-but-great day as we did one of the catacombs (S. Sebastiano), Capo di Bove, Circus of Maxentius, and Church of Domine Quo Vadis on the Appia, and then cut over to Baths of Caracalla, Basilica di Santo Stefano Rotondo, Basilica of San Giovanni in Laterano and Basilica San Clemente before collapsing at the nearest bar. :O)

Thankyou Kathy & Robert.
Great advice!

As Robert mentioned, large parts of the road (yes, it's still an active road!) are restricted to traffic on Sundays so that's a good day to go. We weren't able to do that - I believe we did it on a Friday - but still had a great time; there was very little traffic at the early-morning point where we started, and it didn't get heavier until we were much further along. Narrow dirt paths runs along the paved parts of the road in more rural sections that bikers tend to use and which walkers can jump onto if needing to avoid vehicles (see the pix in the link below.)

https://www.alamy.com/rome-italy-via-appia-antica-appian-way-people-walking-and-riding-bikes-amongst-ancient-roman-funerary-monuments-l-r-primo-monumento-in-lateriz-image412937842.html

I'd strongly recommend doing a virtual walk on Google maps to see what the scenery looks like at any given point. I recall the busiest part starting somewhere about tomb of Cecilia Metella and onwards. You can avoid a busy section by using quieter Ingresso Catacomb di S. Callisto that runs between Catacomb S. Sebatiano and ends near Chiesa Quo Vadis: reference a google map.