Chain length ok?
 

I have a setup of 39/53 in the front and 12-25 in the back with the chain length cut to the specified. Will I have to get a new, longer chain, if I will replace the rear with a 13-28 cassette (front would remain 39/53) or will the chain residing on the bike now be of an adequate length? I have searched, and only found answers for a cassette size 12-27 replacement.

1.5 tooth difference in chain wrap length.
3 possibilities,
prior chain had enough slack that it will be fine
install a half-link
stay out of big/big gear combo

Do NOT rely on doing this. Make sure the chain is long enough to allow big-big or a moment of inattention could have very expensive consequences.

About the only way you could get by safely is if the fractional part of your link was nearly an inch long. IOW, you had something like 1-7/8 link sets slack.
That might be too tight for good shifting, but it should prevent damage if you go BIG:BIG.

That's what I think too. You can ride along avoiding the big/big for a long time. Some sunny day, however, when everything is going wonderfully, you'll try to push up a little hill in the big ring and do $400 or $500 damage to your bike in the blink of an eye.

There are several problems here. You have no idea how the chain length was set, currently. You must have enough length to wrap the big/big. Ideally, you should have enough RD wrap capacity to handle any gear combination. You have three more teeth to wrap, when in the big/big and that requires .75 inch of chain. If the chain length was set using the big/big method, it will be too short.

If the chain has seen much use, then it really should be changed for a new cassette. Then you can use the big/big plus 1 inch method to ensure adequate length. If the chain hangs loose in the 39/13, then the RD has inadequate wrap capacity. If the big/big method is used and the ends of the chain that come together can't be joined (two sets of inner or outer plates) then you must add 1-1/2 inch or three links. This possibility is often overlooked in most chain length setting instructions.

I try to always use a RD with adequate wrap capacity and use the little/little method of setting the chain length. With that method, you make the chain as long as possible, while still maintaining a minimum tension. This method produces the maximum wrap capacity.

This is obviously the ideal way to go. However, if you, as I do, use a triple crank, even a long cage road derailleur, a wide range road cassette and change the original granny for a smaller one you are going to exceed the derailleur's wrap capacity and eliminate the use of small-small. The chain MUST allow big-big for safety reasons. Sacrificing small-small is no problem and inadvertently shifting into it is just noisy, not damaging.

"If the chain hangs loose in the 39/13" don't use that gear combination, you are cross-chained and there are better combinations with nearly the same ratio. A loose chain won't damage your bike, a too-short one will.

I have shortened the quote and I hope you don't mind.

It is good practice to replace the chain when you replace the cassette. In theory at least, depending on how much use the present chain has had, that should result in a longer life for the new cassette, which is the more expensive item of the two, so my point is - why would you not? I am a little surprised that no one else mentioned this, although it is not strictly an answer to the question you asked. I usually pay no more than about $20 for a chain. Cheers.

Thanks to all who took their time to write their suggestions. After weighing things out, it will seem logical for me to purchase a new chain (the cassette is new as well) to be sure of the proper length and to avoid complications.