Your city is overcrowded with cars, (tube) rail infrastructure is not possible.
With limited finances and some piece of the imagination, there could be different solutions.
Very interesting approach is realized in Curitiba (Brazil), they have built their own tube, but instead of trains there is a bus.
Bus stops look like tube station,

The way people get inside the bus, only from these stations (tickets are not sold at bus). You do not need to pay for new ticket, when you are changing the bus on the crossroaded bus stop, this could be even more flexible then european approach (when you have ticket, say for 1 or 2 hours, it varies in different cities). Buses are going each after 90 seconds.
Of course with Latvian streets, the buses like in Curitiba (capacity 270 people) are impossible.
The second problem we have chaotic routes system, it is very difficult to understand the structure of bus routes. Moreover there are different routes that are in fact connected, when part of route is going by the same way of other route, but you are not able to change the bus for free. Mostly all buses are going from the central station to district, but it could be worth to try separate buses into two parts,
– first part that is operating in the center and on the boarders of big districts, these should go more frequently;
– second part, that is operating in the city districts, frequency of the buses there should be connected with the number of people leaving there.
The main problem there are additional resources required to reconsider all that stuff, as well it would be very difficult to change traditional routes and habits (as granny is used to use bus number 3, there will be shock, when bus 3 will not come any more or she will need to change it).