Specifically talking about the Spirit of Fire, but not restricted to that if there's other examples. Can they travel fast enough that time dilation would be noticeable, or would Serina's life be over before they even leave the Shield World's star system? Sure, I know that even if they were going at 50% the speed of light they wouldn't exactly be saying "we're nearly there!" any time soon, but could they even get to that speed in the first place or would they be lucky to get 10%? Or is even 1% out of their reach?
I've been unable to find a canonical top speed for any "fast-mover" in the Halo universe. There's a very brief mention of a railgun (the MAC) that can accelerate objects
"up to 40% of light speed"
in the Halo Encyclopedia as well as a slightly fuller explanation of how the sub-light drive systems actually function, through the use of artificially controlled gravity. This strongly suggests that the upper barrier to sublight is the speed of light itself.
On the assumption that the Spirit is capable of a continual thrust of 1g, per this excellent answer, their speed would top out at just below c after around a year of acceleration, after which they'd simply need to wait until they were a light year from their destination before starting to decelerate. Even if the ship was capable of drastically faster acceleration, the most time they could theoretically shave from their journey is two years.
The unlockable in Halo Wars states that the return journey of the Spirit will take a considerable amount of time. There's no mention of relativity effects but I think we can assume those would take place:
Thanks to Sergeant Forge's sacrifice, the Spirit of Fire escapes from the Forerunner Shield World. Spirit of Fire sets a course for home as the majority of the crew prepare to enter cryo sleep for a journey that will take years, if not decades.
It's established that the Shaw-Fujikawa slip drive can accomplish about 2.625 light-years per day.
As for sublight speeds, the ships depend on power output.
Going off of Halo 4 - Midnight, the Chief had managed to catch up to the Didact's ship despite it having a head start, and at one point Cortana noted the Didact's ship was 300 kilometres away. Several seconds later the Chief had caught up to the Didact's ship and hid under its shields to accompany it in slipspace. Do note that this requires rather extreme accelerations and requires a great amount of force in order to perform this feat. They'd be going at least 216,000 kilometers per hour or over 134,000 miles per hour.
As fast as plot permits.......jokes aside In the series technical aspects, UNSC ships use a Fusion powerplant for energy, also the plasma for this reactor is channeled as a "Fusion drive". Using magnetic acceleration to "shoot" the exhaust particles at high velocity. According to Halo Apocrypha, A frigate using it's MAC gun they can accelerate a solid 600 ton projectile to over 600 kilometers per second. Using the same magnetic accelerators one could excite exhaust particles at similar or superior velocities. In the novel "Fall of Reach" ships at sublight speeds are capable of crossing interplanetary (within solar systems) distances in less than an hour.