The easiest and cheapest "at home" thing to do first is:
Not just change the tranny oil but to flush to whole trans oil system with new fluid and change the trans oil filter
The filter should be located within/underneath the trans oil pan.
If the oil in the trans has discolored and smells burnt this is due to it overheating from excess friction which breaks the oil down.
Oil can only withstand a certain amount of heat before it breaks down and its protective properties diminish.
(think of when you cook with butter and put butter in a pan...on low heat it stays yellow...if it gets to hot it turns
...BROWN)
This heat can also develop if the trans filter from being clogged is not letting enough oil get to the torque converter as well as to the clutch servos which engage and disengage the gears. If the gears are engaging and disengaging due to starving for oil due to a clogged filter not letting oil flow freely...the clutches themselves soon start to glaze which results in slipping.
(if there is oil starvation the trans kicks itself out of gear from loss of hydraulic pressure)
The trans itself only holds a few quarts of fluid while the torque converter hold the majority of fluid often 3 times more than the trans itself. This being the case if the trans oil was drained and only filter changed then buttoned back up...the remaining majority dirty trans oil still in the converter will soon mix with the good oil just put in and you're back to square one.
Hence why its good to start with the system being flushed and a new trans filter, then if any problems persist the next step would be to take it to a garage/mechanic.
-Easiest way to flush the system is to disconnect/unhook one of the trans fluid lines that go to the trans cooler. This can be done at any point in between the trans and the cooler. If there is not a separate cooler often the radiator doubles as the trans cooler. I dont know on your particular vehicle how the system is arranged. Just look for two fluid lines of the same size connected to the trans.
There should be one line coming out of the transmission going to the cooler to cool the heated oil and one coming from the cooler going back to the transmission.
-there is a pump inside the transmission itself which constantly circulates this fluid being pumped out of the trans into the cooler through it and back into the tranny. This happens while it is running even when in park or neutral
-by unhooking one of these lines youve opened(created a break in the circulation of this system. (think of it as simply cutting your garden hose in half right in the middle)
-by opening the circulation of fluid between the trans and cooler you may now flush your trans system by running one end of this line into a drain bucket
-connect the other side of the opened line into a bucket with your clean fresh trans fluid which will be sucked up by the trannys own phenomenal power like magic.
-change your trans filter before continuing to the next step
-now when you start the car the bad fluid will be pumped into one bucket
-and the good fluid will be sucked into the transmission from the other bucket
-continue to do this with the car running until clean fluid starts to come out into the bad fluid bucket
(think of it like if you had a jar with muddy water and you poke a hole in the bottom and keep filling the jar with clean water until clean water comes out)
After this process been completed all the fluid which circulates in the torque converter, servos, gears and clutches will be fresh fluid which will eliminate the possibility of any problems being related to just dirty trans oil.
Therefore if any further problems persist, they may be narrowed down and assessed more effectively.
This has often helped many a car if no other damage has resulted.
A trans flush of this type should be done every 50,000 miles
A synthetic fluid such as Amsoil will resist thermal breakdown and hold its protection and lubricating properties much better than standard petroleum based fluids. Of course this added benefit makes synthetic more expensive but not really considering the fluid holds up better, lasts longer and can help prevent damage which costs money anyway in the long run.
For your scenario of doing a flush...I would use standard cheaper petroleum based fluid until it has been determined if there are any other problems such as possibly a servo having gone bad or a clutch carrier inside the tranny broken due to it popping in and out of gear from lack of fluid or something else.
If your trans works much better and theres no other problems( a certain amount of prayer can help)
I would do the same flush and filter change again within the next 1000 miles filling with synthetic fluid this time.