mode of travel

Train. Just under 3 hours, no change of train. Buy early for cheapest fare - you can easily pay as little as 35 Euros per person. Book on Trenitalia website.

2 hours and 52 minutes, direct trains in the morning from Termini at 6:45, 8:45, 9:45, 10:45. No way way an airplane would be quicker.
Cost depends if you are willing to buy far in advance a nonrefundable, non-changeable ticket. Base price (walk-up fare) is 81 EUR.

We depart in Rome on Friday Sept. 22. our final morning of our 17 day tour. We need to be up to Lake Garda close to Verona that afternoon or so. How much in advance do we need to purchase the cheapest train ticket to Verona? Would we do it now May 2nd. We are thinking that train would be great for viewing the countryside. Also, how do we get to our accommodations on Lake Garda South shore from Verona train station? Thank you. This is our first Rick Steves' trip.

I just looked at mid July and tickets to Verona were as low as 19.90 EUR. They book 120 days in advance.
Where on the south shore? You may change trains at Verona Porta Nuova to a train to Peschiere del Garda or Desenzano.

You can buy as early as 120 days prior to the travel date. Or at the station as late as 5 min before departing. Buying in advance has the advantage of having super Econ or Economy dates available. Be aware of restrictions on those discounted fares. Once s train is booked at those fares, no refunds or changes are allowed.

We need to be up to Lake Garda close to Verona that afternoon or so.
Also, how do we get to our accommodations on Lake Garda South shore from Verona train station?

Where exactly is your final destination? That will determine exactly which station you should buy tickets to, and how you get from that station to your final destination.

How much in advance do we need to purchase the cheapest train ticket to Verona?

The way it works is, each fast train train (such as from Rome to Verona) has a certain number of Super Economy tickets and a certain number of Economy tickets. All the rest are Base fare tickets. Super Economy are not refundable or exchangeable for any reason; if you don't take the EXACT train you booked, the ticket becomes scrap paper. Economy tickets are not refundable, but before travel you can apply their value to a new ticket. Base fare are fully refundable and exchangeable.

Super Economy sell out first, then Economy sell out, then only Base are left. So, if you want to be assured of Super Economy, you buy as soon as you are sure of your plans (if they're not sold out already). There are never more cheap tickets released later - once a fare is gone, only higher fares are left. (It's not like an airfare that can go up or down). If you're comfortable with base fare, you can wait, and can even buy the day of travel (these trains rarely sell out - although it is possible).

Again, whether you should buy your ticket from Rome to Verona or to another station depends on your exact final destination, so let us know before you book any non-refundable tickets.

We will be staying at Moniga del Garda on South West shore for 3 nights (Friday thru Sunday). We have lodging reservations there and joining German friends for two nights. We will be needing to rent a car but unsure at this time for what length of time. Perhaps upon our arrival from Rome, either Verona or ??? If we travel by train from Rome to Verona and decide to rent a car upon arrival where would we pick it up? If we rent it a few days later and are staying at Monica del Garda where would we have to go to rent a car? Still would like to know how we can get from train station in Verona (or wherever else you suggest) to Moniga and our lodging if we do not rent a car. How do we contact a car rental agency? Here in the US? We also have Mon.-Friday of the next week to tour. Undecided where at this time. We fly out of Milan for Portland Oregon on Friday, Sept. 29 and probably stay overnight in Milan on the night of the 28th. Suggestions of places to see during that time would be helpful. Since it is after our 17 day tour of Italy, and after spending 3 nights with friends at Lake Garda, perhaps we would be ready for a more relaxing place where we could just stay put and enjoy outdoors, wine tasting and day hiking plus some sightseeing. Thank you for your help.

There is a Maggiore rental agency right in Verona Porta Nuove station (Verona's main train station) and a Hertz about 1/4 mile away.
Maggiore is a big Italian rental car company and you will have the option of of dropping it airports and major train stations in Italy, as well as city locations. https://www.maggiore.it/en/
The Hertz location looks like a local franchise and you would probably have to return the car there.

You will need a International Drivers Permit in addition to your drivers license. Get one at AAA.

Bone up on ZTL's, Zona Traffico Limitato, in the historic centers of most Italian cities in which driving is forbidden without an advance permit, and how to recognize the signs. They are enforced with traffic cameras and if you venture into one, you will get your credit card dinged by the rental agency every time the police ask "Who rented this car on such and such a time and date". And that is just the fee for telling the police your name and address. The actual tickets will come much later.