flat tyre even after replacing the tube? So I bought a used Specialized Sirrus 2011 model as my first bike (so I'm new here) recently and I love riding it, light weight, smooth handling and comfy to ride. Rear wheel was flat when I bought it so guy gave me some money to replace the tube. 

I took the bike home and removed the tyre (tyre size is 700*28cc), took out the tube and I could see a puncture on the tube when I inflate it. Was quite happy as I found the reason for the flat tyre. I replaced the tube ride for about 30 mins and put it back in the garage. But on the very next day, the tyre was flat again. 

I took the tube out, inflated it but couldn't find a puncture. So I left the inflated tube for few hours and could see that it's deflated a bit. I checked the rim to see if there are any spokes sticking out and no, seems it's all good. Couldn't find anything on the tyre as well.

So anyway, I inflated even more, hoping to find the puncture in the tube. I was squeezing the tube and it got BLASTED!  

Now I'm wondering what to do? Should I try with another tube? Also the given that the rear tyre is quite worn out and could see wear marks (rubber on the top center has brown color here and there), and has fibres coming out from sidewalls. 

If I put a new tyre and a tube, this should solve my issue right? As there are no visible spoke problem with the rim. This's driven me really impatient, what should I do?
Start by replacing your tyre.  If you can see any part of the fibres at all, then it is long overdue replacing!  Buy a new tube too.

Then, inspect the rim and rimtape, if anything looks like it may chafe or cause an issue, then repair / replace it.  

Tubes go flat if left for a long time in any case. If you buy a decent tube, they will keep their pressure for longer. I only buy Schwalbes for my roadie and easily pay more than $10 a tube, but I ride competitively, and I know that they will keep their pressure for at least a month before needing topping up.