When working as the interior finish subcontractor we usually come in after the drywall is finished and painted and we can install the interior doors, window and door trim, Baseboards, and anything else that would go in at that time (Crown moulding, ceiling beams, Stair balustrade, etc.)
Ideally, we want to be brought in near the beginning of the project planning phase. If you know what style of trim you want then we can make sure the doors and doorways are sized appropriately for your finishes. It helps keep you from ending up with casing that has to be ripped down beside doors.
We also want to be able to come in at the end of the framing stage (before drywall) and have the opportunity to finetune the framing details in some key spots, as well as adding the blocking that we need for certain aspects of the job as well.
It depends on your situation and preferences to an extent.
Wood has some advantages that we think are important.
Wood can handle moisture significantly better than MDF. We don't want any of our mouldings to get wet (ideally). It will lead to issues, but the issues when MDF gets wet are catastrophic and unfixable. Because of this we will always install wood in bathrooms or laundry rooms... but if there is room in the budget it is nice to go even further.
Wood is stronger when glued together. We like to glue or screw our window and door trims together so that they won't open up in the future. We want those joints to stay tight for as long as possible. Because of that we would much rather work with wood than MDF.
Wood can hold a crisp edge. For mouldings with delicate details or sharp edges, wood is your only option. MDF mouldings will always have a rounder edge and can look a little "fuzzy" because there are no crisp lines in the profile.
Wood is more durable. In our opinion, wood is usually harder and more durable when it comes to the dings and bangs that normal life inflicts on our homes.
Wood can be "fairly" affordable if you choose a finger-jointed pine or poplar. Still more than MDF, but closer than many people think.
MDF does have some advantages though.
It is more affordable.
It is very smooth. Because there are no knots or wood grain in the product at all, it allows you to end up with a very nice paint finish.
It is also more flexible. So when installing baseboard without a shoe mould, the MDF is easier to bend to match the floor (avoiding extra labour costs scribing the baseboard to the floor). That is true with new houses. If your house has dramatically uneven floors then even MDF will not be flexible enough.
If budget is a concern, but you want the best of both worlds, I often recommend doing the casings (window and door trim) with wood. That allows us to glue and screw them together for maximum durability (reducing joint seperation over time), and gives you a more resilient product around your doors and windows where things often bump into them. Doing the baseboards out of MDF (except for wet areas) then allows you to save some money on the project, without affecting the overall quality as much.
We prefer to cope our baseboards whenever possible. There are a variety of reasons for this, including mainly the fact that is allows us to pressure fit our joints so that they are less likely to open up over time. If the joints do open up, its also less noticeable on a coped joint than on a mitered joint. Feel free to ask us if you have more questions or want a demonstration of what we mean by that!
We sometimes use a shoe-moulding at the bottom of the baseboard. This does several things for us. On a new build house it gives us the option to paint the baseboards on the wall. That means we can install them (about 3/4" off the floor), caulk the top edge, fill the nail holes with filler, sand everything smooth, and then apply paint over all of that. It completely hides the nail holes, hides the caulk, and leaves you with the best possible finish. Then we slide the flooring underneath later, and the smaller shoemould covers the gap and finishes it nicely to the floor.
This gives the additional benefit of also allowing you to change your flooring in the future without removing and redoing all your baseboard. Simply remove the shoemould and replace your flooring, and then tack it back up.
It also adds character to the moulding, allowing for greater depth and detail than you would typically get from a baseboard. But that just depends on whether you like that style, or if you prefer a more minimalist style when it comes to baseboard.
If this is something you would be interested in, make sure to let us or your builder know very early on, as it does change the order of construction slightly.