Three glorious old continetal style cafes worth visiting for cakes and coffee are:
Ruswurm (Buda Side - tucked down street near castle
Gerbaud (Buda side - very polular)
Cafe inside Hotel Astoria
All three are atmospheric, great coffee and cakes and marzipan stuff for those with a sweet tooth
www.ruszwurm.hu
www.gerbeaud.hu
www.danubiushotels.com/en/budapest-hotels/hotel-astoria-budapest
Updates
Gerbaud is on the Pest side (Vörösmarty tér) city centre end-of-the-line on Metro 1
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Another fantastic place to visit is the "Daubner Cukrászda", though it's only take-away. Be prepared to queue up for 15-30 minutes, but their cakes and ice cream are worth the wait!
To get to Daubner take the HÉV from Batthyány tér to Szépvölgyi út and walk up the hill for about 5 mins. it's a one-storey building on the left.