Ayrshire

• is the county town. The main reason to visit is for the birthplace of Robert Burns, in the village of Alloway at the south edge of town.

is where Viking power in Scotland was broken: see the Vikingar exhibition. Largs is the ferry port for Great Cumbrae.

is a small town with a nuclear power plant looming over it.

along with Saltcoats and Stevenston has beaches, but the prime reason to come is for the ferry to Arran.

: Burns once worked here, and there's a maritime museum, but the developers haven't left much of the old town.

is home to the Royal Troon Golf Club, which often hosts the Open.

has a large airport, with few passenger flights nowadays, but which hosts a biannual air show. Plus plenty of golf.

: being the furthest out, it's kept more of its character as a fishing port. Boat trips to Ailsa Craig sail from here.

Arran with its ferry port at is the most attractive part of the entire region.

Great Cumbrae: its town is and a day-trip will be plenty.

Ailsa Craig is the lonely pyramid away to the south. It's an uninhabited wildlife reserve.

has an old church, ruined castle and freshwater loch.

is post-industrial, and frankly there's no reason to visit.

has a ruined abbey and castle.

By plane

is some 30 miles north of the county but usually the best option, because of its wide choice of UK and European destinations. Take the airport bus to Glasgow Buchanan bus station then train or bus into Ayrshire as outlined below.
Prestwick Airport (PIK) is central in Ayrshire and is a major centre for aviation services, air cargo and military flights, but nowadays has few passenger flights. Ryanair is the only commercial carrier, flying to 16 destinations around the Med, Canaries and Poland. So Prestwick sends Glasgow folk off on their holidays but you'd only use it to get in if you were based near one of those destinations, such as Barcelona. PIK has its own railway station and air passengers are eligible for half-price fares on their connecting rail journeys - see Prestwick for how this works.
If you're flying in and renting a car to tour around, then consider also Edinburgh (EDI) and Manchester (MAN) airports.

By train

Wikivoyage has a guide to Rail travel in the United Kingdom.
Railways in Ayrshire radiate out from Glasgow Central station. Trains run every 30 mins through Kilwinning where the line divides: one branch loops back up the coast via Ardrossan and West Kilbride to Largs, while the other branch continues south to Irvine, Troon, Prestwick airport and town, and Ayr. Change at Ayr to continue south to Girvan and Stranraer.
The other line from Glasgow Central runs to Kilmarnock (which also has trains to Ayr and Stranraer) then south through scenic Nithsdale to Dumfries and Carlisle. You can get here from England this way but it's usually quicker to take a fast train from London Euston via Carlisle to Glasgow Central then travel out again to Ayrshire.

By bus

All the main towns have an express bus service at least hourly from Glasgow Buchanan station. The main routes are X34 / 44 to Irvine, X36 to Ardrossan, X76 to Kilmarnock and X77 to Prestwick and Ayr.
Bus 585 connects the coast every 30-60 mins between Ayr, Prestwick town and airport, Troon, Irvine, Ardrossan, Largs and Greenock.

By boat

Ferries to Ireland no longer sail from Ayrshire, you need to go to Cairnryan near Stranraer for the car ferries to Belfast and Larne. A twice-daily bus from Glasgow stops in Ayr on its way to Cairnryan and Belfast.
Ferries from Ardrossan sail to Brodick on the Isle of Arran. You can cross that island by road for the ferry from Lochranza to Claonaig, near Tarbert in Kintyre.
Just north of Ayrshire, frequent ferries sail from Wemyss Bay to Rothesay on the Isle of Bute. You can similarly cross Bute and take a ferry to the Cowal peninsula, then a further ferry onward to Tarbert.
There's also a ferry from Largs to the Isle of Great Cumbrae, but there's no other transport from that island so you have to come back the same way.

The main transport routes within Ayrshire are described above in "Get in". Beyond that, it's variable, see individual towns for local buses.
There's a good road network across the county.

• The Isle of Arran is the most attractive part of the entire county, because of the way pastoral lowland country rubs against stark highland scenery: the glacial valley of Glen Rosa, the screes of Goat Fell, and the lonely "String" and "Ross" roads. Ferries sail every couple of hours from Ardrossan to Arran's main settlement of Brodick, where the castle has re-opened after lengthy refurbishment.
• Alloway, a village that's become a suburb of Ayr, was the birthplace of the poet Robert Burns. You can visit the thatched cottage where he was born, the church and bridge (Brig O'Doon) featured in his ballad Tam O'Shanter, and a museum celebrating his works.
• Culzean Castle (open April-Oct) is a magnificent 18th-century mansion designed by Robert Adam, set in an extensive park 12 miles south of Ayr.
• Other castles: every few miles along the coast you'll find the scrappy ruins of a medieval castle - not much to see inside (even if it's safe to enter - or even possible to distinguish what was once inside or out), you come for the coastal walk and the sea views. Those that are substantial mansions are Culzean (as above), Brodick on Arran, and wacky Kelburn near Largs. Some have been converted to luxury accommodation, such as Law Castle at West Kilbride, and Glenapp south of Girvan.
• Dolphins, porpoises, seals and sea-birds: always be on the look-out for these around the coasts and ferry routes, but the best spot is the nature reserve of Ailsa Craig, reached by boat trip from Girvan.

• Golf: there are dozens of courses, constructed on the sandy machair coastal terrain, see individual town listings. The three standouts are Trump Turnberry north of Girvan, Royal Troon, and the Old Course at Prestwick where the first British Open tournaments were held. There's even a combi-package to stay for a weekend at Turnberry and play all three, a steal at £2000 a couple.
• Cycling: the coast route from Ayr to Girvan has pleasant views, but on the return route the stiff breeze should be behind you.
• Visit a distillery at Lochranza on the north tip of Arran, which produces malt Scotch whisky. Their new facility at Lagg can also be visited but its product is not yet on sale. (Girvan has the big Grant's whisky and Hendrick's gin distilleries but they're not open for tours.)
• Visit a nuclear power station at Hunterston, West Kilbride - you need to book at least 3 weeks in advance.

Events

• The Scottish Grand National is held on Ayr racecourse in April. This course has flat-racing in summer and National Hunt in winter.
• Ayr County Show is in May.
• Ardrossan Highland Games are held in June.
• The Scottish International Airshow is held in alternate years at Prestwick Airport, with the next on 4-6 Sept 2020
• The Viking Festival is held in Largs in the first week of September.

• All the towns have cheap and cheerful places: pizza, kebab, Chinese, tandoori, plus industrial quantities of fish & chips. Don't save it for Burns Night to try haggis, go to the chippy and order.
• The best dining is out of town. Some places that have impressed in the mid-range are Wildings in Maidens near Girvan, Braidwoods near Dalry above Ardrossan, Sorn Inn near Mauchine above Ayr, The Vine in Prestwick, The Catch at Fins in Fairlie midway between Largs and West Kilbride, and Stags Pavilion at Lochranza on Arran. Splurge at Enterkine House near Ayr, or at Trump Turnberry north of Girvan.

All the towns and many villages have pubs. They're quite traditional, serving a local clientele, and there's no outstanding place worth travelling any distance to sup at. JD Wetherspoon are the ubiquitous national chain, open from 8:00 for breakfast, and reliably clean and well-run.
There are a few nightclubs in the larger towns, but Ayrshire doesn't really do nightlife: carousers and stop-outs head into Glasgow.

Standard advice applies about dressing for a change in the weather, not leaving valuables on display in your car, and steer clear of the occasional Friday night drunk.

• You're likely to travel via Glasgow, but don't treat it as just a transport hub. It's a magnificent buzzing city, lots to see do & eat, and it deserves a stay of several days.
• To the south, Stranraer is an eyesore but the Mull of Galloway beyond is attractive. Travel via Cairnryan near Stranraer to reach Belfast.
• Dumfries is a pleasant market town, Burns' later home, and a good base for exploring this rolling lowland region.
• The ferries to Arran create a short-cut to Tarbert on the Kintyre peninsula. Several Hebridean islands can be reached from Tarbert, and more from Oban further north.