I own a single speed MTB . I'm planning to upgrade it to multi speed. As I'm on a tight budget, should I just upgrade my rear to multi speed or upgrade both front and rear. Is it a good idea to upgrade just the rear to multi speed? Edit: My front gear set is a single 38 T one. I'm adding the pictures of current front and rear gear set for more clarity.
Some problems in your plan that I can see from the pictures.
There is no rear derailleur hanger provided on the rear dropout. Without it, you cannot install a rear derailleur.
There seems to be no place to install more than one rear cog, as the rear hub is designed to take single speed BMX-styled freewheel.
Without the possibility to have a derailleur and cassette, you might want to have a rear internally geared hub. This means replacing the whole rear wheel. The new hub has to have the same width as the old one.
To install a front shifter and multiple chainrings, you will have to replace the existing crankset and the single chainring it has with another crankset that can take two or three chainrings. Then, these two/three rings should also be bought and installed. None of existing parts can be reused.
It is not clear from the pictures, but most likely to install the front derailleur you will need an adapter on the seat tube. Even with such an adapter, it is not clear if the front derailleur will be in a right place to work.
A narrower chain is often required if a cassette is used.
A method to draw and attach shifting cables along the frame from shifters to derailleurs or the hub has to be devised.
The list of possible modifications is already so long that implementing them while being on a tight budget is impossible. At this stage, selling this bike and getting another (used) one with gears is indeed the cheapest approach.
my favorite OneBy drivetrain is micro shift's adventx.
it has a wide raito for a 10spd and is on par with shift quality of Sram's GX 12spd (it comes close to the same gear ratio to)
also cluches are important and this one is not as effective as Shimanos shadow, but better than Sram's paw and ratchet clutch.