I will be travelling to Gothenburg via Copenhagen. I have a traveller's card that has Euros on it. Will I be able to use the traveller's card to withdraw Euros from ATMs (I have used this card to withdraw Euros in France) in Gothenburg and in Copenhagen? And then how easy is conversion from Euro to Krona in Sweden? None of the agencies I've contacted in India are able to provide me a card loaded with Krona. And I'm hesitant to carry everything in cash.
Not sure about Gothenburg, but I've been in Sweden many many times, so I can comment on the general issue.
The local exchange offices in Sweden (I used the ones in Stockholm and Malmö) charge you a huge amount of commission/service fee. That's far above the standards of any country I've ever been to outside Northern Europe. Considering this, I would really not recommend carrying Euro with you in hope to change it to Krona later on. Same problem appears even stronger if you somehow manage to withdraw Euro from an ATM in Sweden, I'm skipping that option altogether.
Paying in Euro is possible at very few places. I remember McDonalds in Stockholm accepting Euro, their (so to say) exchange rates were not great, but certainly better than what Forex (the local exchange office I used) ends up charging you.
The final two options at hand are using your card either directly or to withdraw local currency. While the choice between these depends on your bank's fees, mine has a policy that every time you withdraw cash abroad in another currency (we use Euro, so this means outside Euro-zone), apart from using their own spread for buying and selling the currency, they charge you a fixed 2.50 euro plus 2.50% of the amount of money you withdraw. That's also pretty hefty, especially if you're withdrawing a relatively small amount.
On the other hand, if you just use your card to pay directly, you only lose the difference between the real rate and the rate that the bank sells that currency. This is by far the best option of those that I've counted so far. The problem with this one is that you cannot purely rely on your card; it's not valid everywhere and some places really require that you carry some cash. On top of that, in many countries I would be reluctant to use my card everywhere for safety reasons, but you don't have to worry about it in case of Sweden. Besides, in Northern Europe card payments are very widely accepted compared to most other parts of the world.
So my humble suggestion, withdraw some cash in the local currency, but not much, then pay for the rest of the stuff directly with your card whenever possible in case your bank's fees are similar to mine. If your bank does not differentiate between cash withdrawal and direct payment, you may prefer to withdraw more local currency from an ATM. Have a nice journey!
You won't be able to withdraw Euros easily. There could be a limited number of special-purpose ATM in larger banks but I have personally never seen one in Sweden and/or Denmark (I have seen some in Switzerland). Regular ATM won't offer Euros.
What you should be able to do is withdraw Swedish Krona with your Euro-denominated card (or perhaps even directly use your card in shops). Obviously, someone (card-issuer or ATM operator) is going to charge you for the pleasure, in the form of an explicit fee or a buy-sell spread compared to the current exchange rate but that might still be better than changing a lot of cash in advance and carrying it around.
You could also come with some Euros you bought elsewhere and some businesses will probably accept them but buying foreign cash at home and then changing it once more or spending it at an unfavorable exchange rate is very unlikely to be cheaper than using your card.
Although many businesses in Göteborg accept Euros in cash, the most convenient option is just to use your card for everything. Parts of the transport system don't accept cash anyway. It doesn't matter what currency your card holds; it will be converted to SEK when you make a transaction.
Important to bear in mind: Although some businesses (mostly in hospitality) still allow you to sign receipts instead of using your PIN, they do not check for a signature on the reverse of your card and may instead ask to see photo identification, which will need to be a passport or driving license.
Swedish stores have no obligation to accept Euro or any other currency besides Swedish krona. The exchange rate at the stores/chains that accepts Euro is far from the best for you as a customer.
Your probably best option is to withdraw local currency/pay directly with card (I see other answers already touched upon that). If you want to exchange some cash at a currency-exchange-office there are several locations in Gothenburg, split between Forex and X-change but the fee for exchanging to krona (without a receipt) is a bit high for making several exchanges.
Lastly a bit of advice regarding the public transport system in Gothenburg: Västtrafik (http://www.vasttrafik.se/#!/en/). The systems used to pay the fare is not optimised for visitors: there is no way to pay the driver directly (unless you travel far out from Gothenburg), there is no ticket vending machines at stations, only some vehicles offer ticket vending machines, and to pay via sms you have to pre-register at the website mentioned. So I would recommend either register in advance so you can pay via sms or purchasing a suitable ticket-card from any shop that carries them (marked with the logo) before your first travel in Gothenburg.