Holy - wipe it down with Methyl Hydrate (alcohol) - which you can get anywhere that sells paint.

Prime with one coat of Glidden Ultra-Hide Gripper.

Wait overnight - then topcoat with your choice of color - Glidden Evermore Satin or Semi-Gloss - Glidden Ultra-Hide Semi-Gloss - Ralph Lauren Satin or Semi-Gloss. I would recommend two topcoats.

All are lates products available at The Home Depot

Mr. Paint

In all fairness to Mr. Paint not everyone is near a Home Depot and as such we need to reccomend products that can be purchased others places. With that said here is a way to go.

You will need to sand the trim to give thwe primer something to stick to and this a good practise any time you paint over glossy slick paint. Then you can wipe off with denautered alcohol to remove oils and dirt that get on the surface. Then you can prime with either an acrylic bonding primer such as Zinsser 123 or an oil based undercoater from the paint supplier that makes the finish paint. Then you can finish with an acrylic latex semi-gloss or gloss depending on the look that you want and yes 2 coats will be best.

Happy Painting

5.......I live in the middle of nowhere and I am 15 minutes from a Depot.

As I was just saying in another post - I recommend THD a lot - not because I love THD - but because they are the most common supplier of Glidden products - which I happen to know and like.

If you have an ICI Paints Store - a Devoe Paints Dealer - or if you are in New England - and Aubuchon - or just a mom and pop paint store - you should be able to find Glidden ICI products as well.

THD is just the most likely place for everyone everywhere to find the products I recommend.

Having said that - B-I-N 123 is a good product (Gripper is better in my opinion) - I used it a great deal before Gripper was introduced - and it too will work to prime old oil.

Mr. Paint

ALWAYS use the primer recommended by the paint you are using over it. Paint suppliers know more than anyone else on how to use their products, and they put the instructions right on the can. Zinsser, Kilz, etc. make good products, but not as good as the actual paint suppliers.

I disagree Zinser and such make a really great primer,this is why it is used for some of the tougher jobs.
5 I agree too,this is why I recomend places like H irshfields (I think Mr.) said he has never heard of this.Also Sher win Williams,M enards,and L owes,I use the spaces like this so it doesn't delete what im typing this board is very biased only liking 1 store,all others mentioned get erased.
Anyhow I would sand it like 5 said,this way you know you are getting the best adhesion possible for your primer to stick to.Sanding will also let you use most any interior latex primer you want to use then being able to put any brand of interior paint on top of this.

All this back and forth. Firstly, don't trust the opinions of store clerks. If you want to know a products do's and don'ts call the products tech rep. Secondly, Gripper and Zin123 work fine over oil. You may also purchase a product to mix directly in to latex piant called OIL BOND, no primer or heavy sanding nessesscary. Also If you properly sand and clean the oil surface the latex WILL stick WITHOUT primer or oil bond...but it's VERY time consuming to do it right. If you are still worried about adhesion and and want something with a stronger bond you will have to step out of water based primers. Zinnser Cover stain makes a very strong bond and can be painted over with latex. If you want an INSANE bond you can use Zinnser BIN..I really only use that painting over wood cabinets or other finished wood. In a nutshell, you have serval options:
1. Rough sanding prior to Latex
2. Light sanding a use a water based primer 123 or Gripper.
3. Use oil bond to mix directly in the paint.
4. Light sand apply oil based primer for very strong bond (I recommend Cover Stain).
5. Light sand apply shellac primer (Zinnser BIN) EXTREME bond.(expensive and shouldn't need to go that far unless you painting over something that is in high use like cabinets..or you want to sleep knowing that not even a semi truck running it over can peel the paint off your trim.)