This is probably not so hard as you think, and the natural answer would lead to the development of the 'right' form of space propulsion technology insofar as it would be scalable beyond interplanetary flight full on into interstellar. That said, it's not all as simple as it sounds in the OP. There is afatal flaw in your original assumption.
It's easy to imagine how would a water-breathing aquatic race outfit its spaceship
This actually isn't the case. Air breathers have the benefit of needing a gaseous atmosphere, and that atmosphere has one crucial advantage over liquids; it's compressible. This allows the body to take a much higher level of G force in that atmosphere because the gas doesn't actually add to the problem (actually it does (Boyle's Law) but not in quantities that significantly contribute to the problem).
As an experiment, put a hamster in a cage and shake it. Check your hamster; it'll be bruised, sore and more than a little angry, but it'll most likely be alive. On the other hand, put a lid on a fishbowl containing a goldfish, and then shake it.
There's a VERY good chance your fish will be dead.
(When I say 'Experiment' by the way, I'm not actually recommending you do this. Please don't try this at home and if curiosity persists, please consult your mathematician.)
The reason for this is that water isn't compressible and therefore transfers the full brunt of all its kinetic energy into whatever IS compressible in its path, and in this case it's the goldfish. Water is also very dense, meaning that the mass striking the goldfish with every shockwave of the shake is phenomenal by comparison to its normal environment. The goldfish won't survive.
You can actually remove a cork from a wine bottle by repeatedly knocking the base of it against a wall or a tree. I don't recommend this either because it bruises the wine, but my point is that any living thing taking massive G force stresses in a liquid is unlikely to survive.
Even your Merfolk will be in the same boat, but the answer (provided you can get out of the planetary gravity well somehow) is simple; some form of constant acceleration engine.
NASA is already working on two possibilities in this space; Ion and Plasma engines. These are designed to provide constant, relatively gentle thrust by comparison to conventional chemical rockets. These would be far more efficient for long distance travel than chemical rockets (although they're not powerful enough to get out of the gravity well) and the constant thrust ALSO provides for that simple answer for your merfolk; gravity.
Well, not really gravity, but the constant acceleration is going to gently push everything to the back of the spaceship. This will mean that all the water is going to drift to the back of the ship. If the ship's internal chamber is only 80% water, then the front of the spaceship is 'up', and your merfolk swim that way to breathe. You recycle the atmosphere at the tip of the ship, and the water gets filtrated et al at the bottom.
You will probably need space suits of some form for those short term attitude changes or at the half way point when you turn the ship around and start to decelerate, but for the vast majority of the trip, you have a natural chamber of air at the 'top' of the ship for your merfolk.
You still have the problem of getting out of the gravity well, but that could be done with a space elevator under the circumstances. So; you have a space elevator to get you out of the gravity well, then a continuous thrust engine to provide light gravity and orient your ship. For exploration, this is perfect. For combat, you're in trouble because rapid changes in velocity or impacts on the side of the ships don't have to breach your ship to be fatal. But, that's the price of being a water borne species I'm afraid.