This is speculative. There's a fair bit of engineering to be done for some of this, but all are completely possible using today's technology--albeit expensive. There are no breakthroughs required, just the will to do it. In several cases, they are extremely cheap to operate, while being expensive to initially build.
For the longest, fastest trips, we will use Evacuated Tube Transit, or VacTrains. Traveling in evacuated tubes at thousands kilometers per hour, it takes so long to accelerate and decelerate that it's impractical to share tubes between different routes in most cases, even when they're roughly co-linear. Thus these will be largely point to point routes, initially between our biggest cities, passing under oceans, over or through mountains. There are a lot of difficult engineering problems still to be solved, but the advantages are so large it's hard to imagine this technology won't eventually be adopted. Trips will be in relatively small vehicles, unscheduled, provided as demand requires. Energy use is extremely low--there's no air drag, and rolling friction is negligible (they'll probably use maglev). Once the thing up to speed, it's traveling on momentum until it's time to slow down--and you can use slowing down to generate power! Imagine: New York to Sydney in an hour, leaving any time you wish.
Intermediate distance travel will be implemented with High Speed Rail which resembles the current form, possibly using maglev, traveling at a few hundred kilometers per hour between cities no more than a few hundred kilometers apart and probably no less than 50. It's almost as expensive as VacTrain to build but because the trains are atmospheric they can't run more than about 500kph. Because there's a power and aerodynamic advantage to it, trains will be long and run on schedules. All of today's HSRs are electric and I see no reason to think that might change.
Conventional Aviation will serve routes that are not yet served by VacTrain or HSR. Over the long term, these will more resemble regional airlines than the majors. Aviation has amongst the highest energy demands of all and will thus remain one of the few users of Fossil Fuels and their artificial replacements.
Ballistic travel is possible. The technology used to put people into low earth orbit and return them safely can also be used to provide point to point travel anywhere in the world at speed as high or higher than evactuated tubes, and using it that way will allow economies of scale to reduce the cost and increase the safety for both types of destination. Rockets consume a lot of fuel although some use non-fossil varieties. I think it is really only practical if there's also a significant business putting people up into low earth orbit and recovering them. I think this can only happen if there are meaningful (read: profitable) space colonies, probably mining asteroids.
Conventional interurban or commuter rail will serve high volume corridors that are too short for HSR or VacTrain. Some existing rail is powered by Diesel, but much is electric and over time I'd expect these corridors to be electrified (or re-electrified).
Personal Rapid Transit will largely replace municipal bus, light rail and subway systems. Where practical and grade separated, this will occupy the rights of way those systems used, with a cheaper, faster, more convenient alternative. You can get four PRT lines into the space occupied by a single subway track, and since most of these are two way, that means as many as 8 PRT lines. Since each of these has roughly half the capacity of the subway, that means a net doubling of line capacity. It's far more than is necessary in most places. Building new PRT routes will be far cheaper than anything except bus routes and they are easy to connect up with existing PRT routes. PRT can serve some intercity routes, but probably no more than 50km or so.
Battery electric cars will completely replace carbon combustion cars. These are practical for short routes that are not yet PRT-ized (or so lightly traveled that they will never be PRT-ized). A few examples: very low density dwelling--less than two or three people per acre. Point to point service for handicapped. (PRT is much more amenable for handicapped than conventional autos but there are some situations that just can't be helped). There will be plenty of ambulances and other rescue vehicles serving the PRT, but some need to serve other houses.
A bus is just a big car that carries a lot of people and runs on a scheduled route. Routes that are not PRT-ized, such as low density rural areas, may have a battery or fuel-cell electric bus on it, very much as today. Buses are much more amenable to fuel-cell energy density issues than are cars.
The biggest present problem with battery electric cars is that they can't be used for longer trips. The solution to this is better roads: Provide a charger that travels along with the car as it drives on a longer trip. This probably consists of a dedicated lane on certain roads (I think the present US interstate system is a good place to start) which has some sort of power lanes. When you drive on it, you surrender control of your car to the computer and it couples to the power system and recharges your battery as you drive, billing you by a system resembling the current RFID passes used for Toll Roads and Bridges in some places. Because the computer is driving, and traffic is grade separated from human controlled traffic, the speed limit can be whatever is practical for the road, unlimited by driver limits, and the cars can drive close enough to the one ahead to take advantage of its aerodynamic draft without risking the driver ahead doing something unexpected. When you want to get off, you press the button and the takes takes you off the special lane and merges you with the human-controlled traffic, and hands over control. Travel by car between most urban areas will be a lot slower and less convenient than vactrain or HSR, but there are a lot of reasons you might want to drive--sightseeing, for example.
Freight can be moved by many of these same systems. Most larger facilities will have their own PRT freight dock. PRT freight can intermingle with human traffic up to the capacity of the system. PRT freight can be unscheduled, providing Just In Time delivery for almost all goods. This actually reduces the size of loading docks, while increasing their capacity quite dramatically. I think there's an application for a Less than Car Load transfer points: a container drop off point which transfers directly to PRT freight vehicles without going inside a building. The container serves as a sort of temporary warehouse.
Smaller buildings won't have their own PRT station, so if you need a large item delivered--a refrigerator for example, this will still use a delivery van. Except for being battery-electric powered, this will be just like today's vans. Small deliveries will use couriers riding in the PRT and walking the last few hundred meters, possibly with a cart.
The other modes can also ship freight, although it'll be more like air freight. If it's heavy and large, it's probably better to use conventional containers, on trains and ships. Most of these can be electrified, so they can get their power from renewables, but there's little application for shipping a sofa halfway around the world at 5000kph, although there clearly is a market for documents and other small objects. Long Haul Trucks and Containers should mostly be shipped by train, although something resembling the current long haul truck system can be electrified using the power rails for electric cars. This presents a special wear problem for the dedicated lanes that needs to be resolved.
Petroleum, natural gas, and artificial fossil fuel like fischer-tropsch or algae or ethanol, will still be used for airplanes, and probably some other specialized vehicles needing high energy density (I'm thinking of a pumper fire truck) but it will probably cost $50/gal or more and will only be used where there really is no alternative. Most human and freight transport will be done in vehicles that get all the power they need from their guideway. Battery electric vehicles have a range of 70 miles today and with frequent charging stations and charge-while-you-drive solutions, their effective range can be infinite.
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