Congratulations on your new property! Keeping horses at home is a lot of work, but very rewarding. Here are a few things to look for as you get started.
With regards to the fencing, I would take down all the barbed wire and start from scratch. Check the posts and ensure they are solid, and replace any that are loose or rotted. Good quality electric fencing in a rope or braid is very economical, safe, easy to install and it looks good. Use good insulators (preferably screw in) and be sure to ground the fence properly. I use 4 strands of electric rope with the top and third strand electrified. Alternatively, there are also lots of plastic coated high tensile wires and polymer strap fencing materials available that look good but are a little more expensive. If using any of these be sure to follow the manufacturers recommendations about bracing and tightening. If you are putting in new posts, consider putting in a heavy hitching post where you can tie up to saddle, wash, etc. and possibly cross fencing a "sacrifice" area where you can keep a horse contained if injured and prevent the pasture from being trampled too much in the rain. Make sure you have gates large enough to get a horse trailer / tractor / hay truck into your pasture. Usually 10-12' is good. It is worth spending a little extra on quality metal gates that can be safely secured and locked if needed.
Check the pasture for any debris (scraps of metal, nails, sharp sticks, etc, noxious weeds or holes and remove these. Depending upon the quality of the plants in the pastures, a light seeding and / or harrowing may not be a bad idea in the spring.
You will need a good quality water trough somewhere that doesn't get too muddy with access to a faucet or hose. If it freezes in your area, you will need some sort of heater for winter.
Check your sheds for any nails, sharp boards, sharp edges, etc. and fix these. If they have metal roofs, trim / secure the edges with wood so the metal isn't sticking out. Three sided sheds are ideal shelters for horses as they offer both good protection and the ability for horses to move around easily. Unless you have year round pasture, you will need a secure place to store hay and feed. Perhaps you could fence off one of the sheds for this purpose?
Anyways, these are a few of the major things I would consider right now. If you are interested in showing, it would also be worth finding a good trainer who you could take some lessons from and could give you some good advice on finding the perfect horse for your needs. Congratulations again! Let us know how it goes.