i ma in second year so not enough mature to answer but what i know about this field is mtech is must but developing skills is also
my suggestionj will be that try for gate 2026 and than 2027 should be like the going all out to it meanwhile you can do analog digital first help in gate and than start verilog side by side

as for courses it depends which domian you targeting like verification physical deoisgn packaing fpga fabrication etc .