The question seems a bit vague, how long has the ship been sat there? Have you just joined with all the crew or are you relieving the previous 2nd engineer?
These factors will decide how the question is answered, and just how much work is involved.
Essentially you need to look at everything that could give you grief, or that could get you detained at a future port. Think of it as doing your own port state or USCG inspection. In reality you would judge what is necessary by the state of the ship when joining.
But working towards a worst case scenario
If you have access to the internet before you join you can check the vessel for any history of detention etc... check for previous names and check detention history of these as well - vessel could have recently changed owners and/or names.
You should satisfy yourself that all the emergency equipment is working as required;
Main and emergency fire pumps - ready to run and a hose test to check pressure and power from each pump. Suggest one hose at the furthest hydrant from the pump and one from the highest point simultaineously.
Emergency bilge suction - valve is free and that bilges are clear of anything that could block the valve or the pump.
Machinery space bilge alarms are working
Emergency D/alt starts and can take the expected load.
Emergency batteries are fully charged and will start/run what is required.
Emergency air compressor arrangement is capable of filling the bottle.
Fire alarm system is working and that nothing is inhibited.
Fixed firefighting system is operational and ready for use.
All portable fire fighting equipment is in place and ready for use.
Lifeboats and engines are in good working order.
Sound powered phones at each locations to every other location.
You should satisfy yourself that all the machinery is in good working order;
LO analysis tests of Main Engine (including camshaft system if B&W) steering gear, stern tube, auxiliary engines and may be a few more depending upon the ship in question. If this is not possible there should be equipment onboard to do a water in oil test, samples should at least be drawn to visually check oil condition.
Is the main engine scavenge space clean?
Any white metal lying at the bottom of the main engine sump?
If the fuel has been onboard for a while fuel analysis may also be a sensible test to have carried out.
Auxiliary engines will autostart and that they can take the load expected.
Bilges are oil free and that there are no machinery leaks (oil or water).
Spare parts on board for essential equipment, main engine, auxiliary engines, fire pumps etc...
Essential tools for overhaul or removal of main engine cylinder unit.
OWS is operational and that the bilge and sludge tanks are at acceptable levels.
Fuel and LO purifiers are working correctly and not producing too much sludge.
Alarm system is working and alarms have not be inhibited.
Does the engine room operate UMS? Test the ER dead man alarm and cabin alarms.
"Engineers call" does this work.
Talk to the people you would be working with in the engine room department, ask them about their experience either on the present vessel or previous - you can soon judge their competence levels - and ability to speak the working language.
Familiarise yourself with the emergency controls for the main engine, steering gear, lifeboats, lifeboat engines, and various means of starting the emergency d/alt. Change over proceedures for engine control - bridge to engine control room, engine control room to local and back again. Check the emergency escapes and where they come out, making sure doors are not locked or blocked. Make sure the rest of the engine room staff know this too.
For my own satisfaction I would take a walk along the jetty looking for any large indents, and oil leaks from stern tube and, if fitted, thrusters or stabilisers. Walk along the main deck checking the condition of any cranes or cargo gear and the hatches and covers. Also check funnel for smoke quantity and colour.
You can start the engine and test it ahead and astern whilst alongside, you will need to stop cargo and raise the gangway. It may be necessary to place additional mooring lines out. But a start test of the engine takes roughly 30 - 40 seconds and as soon as it can be seen that the governor is controlling the rpm the engine can be stopped. If the engine is tested in the reverse direction for a similar period the ship should end up where it started
You should test from all control stations, bridge, engine control room and local control position.
If the vessel is fitted with Bridge control and it is proved ok at the start ahead and astern tests then there should be no trouble to give them control. If there is any doubt leave on engine control room control and change over once in clear waters.
You could check discharge books and certificates of the other engineers onboard if the Master will let you.
I would ask to see the latest class status report and check for any conditions of class and any overdue survey work.
You could ask to sight all the statutory certificates;
Dangerous Goods (if applicable)
Vessels maintainence history, defects list and outstanding work list.
Main engine parts running hours
Last main engine bearing clearances and crankshaft deflections.
Outstanding orders for machinery parts.
Read through the engine room log book, and check that all parameters are what you would consider normal.
If possible read through past Chief, third and fourth engineers handover notes.
Who is the designated person ashore, who is the emergency contact?
Any parts of the ISPS plan that you are allowed to see, and your role/responsibility onboard.
Muster list and your duties in an emergency
I'm sure I will think of more as the day goes on, and that the others in the forum will also add some relevent points or comments to the above.
Steering gear and main engine points I will reply with separately, when the boss is not around!