Rome was famously "not built in a day" but you can make a good tour of it in eight hours. Coming from Tuscany and Umbria, most will arrive at the main rail station "Termini".
There is a choice of following our routing, given below, for which you will have to travel by metro, bus and/or taxi and do a bit of walking or simply buy a ticket for the red tour bus. This open topped bus, travels a route around Rome that passes or goes close to most of the places you will want to see in a day. There are buses every fifteen minutes and the ticket (€15) entitles you to get on and off as many times as like. If you do this, make sure you get off the bus to see the Piazza Navone, Parthenon, Piazza di Spagna, Trevi fountains at the very least as the bus can't get to them.

You buy the ticket in the bus station outside the rail station - the ticket office is on island E and  the red buses stop on the far right of the bus station as you look from the rail station. There are also special bus tours starting here for "Christian Rome".  DO NOT take the alternative circular bus tour run by CLT - they are unreliable.

The Pugnano Alto Tour
The route we take with our friends.
Take early train, about 06.50,  from Terontola arriving Rome about  08.30  (There is a later train at 08.11 but it takes longer to get to Rome and may mean curtailing the tour a little.)
Proceed to metro- directly below the main station and buy a book of tickets that can be used on metro and buses (at least three per person). (You can buy them near the metro and they are also on sale at tabacchi shops (the ones with a big "T" sign) You are expected to cancel each ticket yourself - at the station before you start a journey ("validita") and on the bus.
Metro to Ottaviano - walk to Vatican museum (closed Mondays). Its four to five hundred metres. Buy ticket and make way to Sistine chapel. Resist temptation to look at anything else though we usually stop in the medievel maps room.
Walk round Vatican walls to St Peter's square (can send a postcard with Vatican postage stamp). On Wednsday mornings the Pope has a public audience in the piazza (Dilys was once asked by a tourist - "who's that man in white") Bus 64 (possibly other numbers too) from Piazza San Pietro to half way down Corso Vittorio Emmanuele, if possible get off at the Museo di Roma (the bus crosses the Tiber and does a one way circuit before driving down the Corso V.E. get off at the 2nd stop, Museo di Roma);  or take taxi - to Piazza Navone (square with three fountains).
 

From the Corso Vittorio Emmanuele bus stop coming from the Vatican you need to cross the road and follow pedestrian signs to Piazza Navone. The Piazza Navone is the square with three fountains and a world famous gelateria (ice cream cafe). In the 19th century, the piazza used to be flooded so that naval battles could be re-enacted. (A friend says this is wrong - it was in Roman times.)Walk to Pantheon (follow pedestrian signs)- largest extant Roman building - this is the highlight for me though its a pity the old Roman gods aren't still in place.
 Walk to Trevi fountains (all these walks are signposted). This part of the walk takes at least fifteen minutes. Use a map to walk to Spanish steps though there are signs. Rest on the steps awhile.

 Walk down main shopping street opposite Steps (Via Condotti) Catch Taxi to Piazza di Campo dei Fiori, It can be done by bus but I think you would need to change at Piazza Venezia. (Just behind the Campo dei Fiori  is the Palazzo Farnese, now the french embassy, where the second act of Tosca is set, on the bus from the Vatican you have passed Castel Sant'Angelo where the last act is set) Have al fresco lunch in piazza. (important to get here early as the places fill up)

Walk through the narrow streets starting on the via Giubbonari (the one on the top left if you have your back to the via Vitt. Emman.) to come out at the piazza Venezia which (with great care) you cross and climb up the steps to the Capitoline Hill (not the steps on the wedding cake but the ones after them) admire statues and the architecture of Michelangelo - continue through the upper piazza to walk down street overlooking the Forum - take steps down into Forum (closed one day a week).

Check out Forum and leave by Colosseum gate, opposite end to Capitoline Hill. (Don't miss the senate building - at the capitoline hill end); I'm not sure this is the senate building where Julius was assassinated but you can easily imagine it was.)

Colosseum.
Colosseum station for metro back to main rail station Termini.
It's a full day but quite feasible. If you find the walking too much then taxi or bus the route between the Pantheon (means going back to the Corso VE) and Trevi fountain. You could also skip the walk between Campo di Piazza dei Fiori and the Forum. Lunch at the Spanish steps (not Macdonalds, please) and get a taxi direct to Piazza Venezia or the Forum (Foro Romano).

Not recommended for non enthusiasts in August.