Taxis, trishaws and local transport in Burma
As in any country, you will be faced with several means of moving from one point to another locally. If we find classic taxis among the four-wheeled vehicles, generally in the big cities like Yangon, Mandalay, Pathein, Mawlamyine or Taunggyi, you will also find a network of public buses, along the main axes.
Regarding these city buses, the fares are low, at 50 or 100 kyat, yes yes, you read correctly, that's barely 6 cents! This is in any case the case in Yangon with the city's bus system, YBS (Yangon Bus Service). Good news in addition, since 2017, they offer an app available in English on Android.
Convenient to find your way around, knowing that the buses serve the city center and the suburbs, so an extensive network. You have all the information on the different lines, color code etc. on the myanmore site. As a tip, have change, because with large sums it could be problematic…

City bus in Yangon.
Regarding taxis, they will be significantly more expensive (compared to the bus) especially given the tendency to gouge tourists... Because unless there is a change since my last visit, and in any case in Yangon, taxis do not have meters, so they can give you any price...
But there is a trend to be aware of, based in particular on the reference site for prices abroad (Numbeo), the base price should be around K1 – K000, then expect about the same per kilometer. So if you are going to the city center from Rangon airport, which is about 1 km, you should be looking at something between K500 and K15, or around €15-000.
If you want to hire a taxi for a day, expect to pay around K5 per hour.

Taxis on the streets of Rangoon (Yangon).
Then, similar to Thailand, we find pick-ups that make intermediate trips, sometimes around cities. These are fitted with three rows. For short trips it can work, but for longer ones, the comfort/price ratio will not be worth that of local buses. These pick-ups are often crowded, and the only way to avoid too much hassle is to reserve the seats at the front, more comfortable, but therefore negotiable at 25% or even 50% more expensive than those crammed in behind.
They make connections between cities and intermediate places, generally making numerous stops to load/unload. They are thus found between Bagan and Mount Popa, or, not far from there, between Myingyan and Meiktila (which brings you closer to Kalaw and its trek departure point) or around the Golden Rock in Kyaiktiyo to name just these examples.
After the transport par excellence in Burma, there are motorcycles, fitted with sidecars, which are locally called "saiq-ka" but whose generic term is rather trishaw. There will sometimes be simple motorcycle taxis, as we could find in Bago from the bus stop and which we took to our hotel. We also find its penchant without an engine, but clearly more intended for the most deprived locals.
More rarely you will come across types of tuk-tuk similar to the Thai or Indian neighbors. A trip costs on average 4 – 000 kyats depending on the distance (€7 – €000). Also quite common are the old-fashioned horse-drawn carriages or even carts pulled by oxen (rarer though).

Short ride on a motorbike taxi when arriving in Bago.
Car, motorcycle and bicycle rentals
To date, it is not not possible to rent a car private for oneself, or at least it is restrictive. Basically you will often read that it is necessary to rent a vehicle with a driver. The country is not yet "open" enough to let foreigners roam freely on their roads. You will find for this various agencies allowing a rental by the day or even for several days. After all, it is not cheap. A round trip between Yangon and Bago for example is priced at $110 on this site.
But recently, it has become possible to drive yourself. To do so, you must obtain authorization from the government agency MTT and the Road Transport Administration Department (RTAD; 01-36113). However, the constraint is that you must be accompanied at all times by a local... so a guide in short, because otherwise I can't see how you would take a random guy as a "guarantee"... Basically, these restrictions are there to prevent the government from having to deal with accidents involving foreigners.
On the other hand, it is possible in some places to rent a motorbike or scooters, generally outside the big cities, even if it is a priori possible to find some in Mandalay. This is what we did during a memorable day in Chaung Tha, between the beaches. We managed to find a bike for 10 K per day (€000), the usual rate you should find. Gasoline costs next to nothing and you will find bottles selling it by the liter almost everywhere on the side of the road (normally 650 K per liter, or barely 40 cents).
We also rented a scooter to visit the surroundings of Hpa An, a great region to see and ideal to explore in this way.
Concerning Bicycles are much more widespread and cheap. Prices are around 1 K per day For a basic bike, it can go up to 4K if it is offered by a high-end hotel. Always make sure to check the bases, brakes that work, tires inflated correctly to avoid any problems.
You could even buy them on site, you can find bikes generally coming from India, China or even Thailand, sold for barely €100.
If you are a long-distance traveler and with your own mountain bike (preferable in Myanmar), you should not have any problems with customs, and traveling within the country (outside of restricted areas) is not a particular problem (there is little traffic outside the cities). If you are looking for a company offering bike rentals, sales or even bike tours, you have Cycling Myanmar, based in Yangon: https://www.cyclingmyanmar.com/
Another possibility, still in Yangon, the company Thin Asia Tours: https://think-asia-tours.com/