Analyzing a circuit with 2 voltage sources
I first thought to combine the voltage from both sources and use it to
calculate the current and voltage across R3 but then I get stuck with
what to do for R4 and R5.

Well, for a start we talk about through and not across. Secondly, the current in R4 will always be the same as the current in R2 so, because of that, R4 can be moved from where it is to be in series directly with R2. Combine those two resistors into one value and you've immediately simplified the problem to make it easier to solve. So knw you can....

combine the voltage from both sources and use it to calculate the
current and voltage across R

But, be careful here because you can't simply add the voltages; you have to combine them with a technique embodied in Millman's Theorem: -

And take special care of R3 because it is in series with 0 volts.
Of course you could convert the simplified circuit from voltage sources to current sources and then add the currents but, that's what Millman's does anyway.