The main tourist season is from April to October, with August being the busiest month for the usual reasons (book ahead, or avoid it if you can). Savvy travellers tend to choose April-May or September-October, which are quieter but still warm. We braved November, and loved it: few visitors, mild days and a top-up of prewinter sun. Temperatures don't vary much throughout the year, averaging 17-20C in January-February and 25-27C in July-August: what geographers call a temperate, subtropical climate. Be prepared for rain at any time of year - clouds roll in from the Atlantic and condense especially on the higher summits - as well as strong winds in autumn (September-November). April-June is generally best for whale-watching, as you may spot migrating blue or fin whales.
From UK: Ryanair and Easyjet fly directly from London (mostly Stansted) to Ponta Delgada on São Miguel and, less frequently, to Terceira. BA flies from LHR to Ponta Delgada, excluding winter. Journey time is 4 hrs 30 mins. Otherwise you'll probably fly via Lisbon or Porto - TAP can do a combined ticket - and both legs take 2.5-3 hours.
From Europe: Ryanair and Easyjet fly directly from major cities such as Munich & Frankfurt; there are also seasonal direct flights from Amsterdam, Geneva, and Brussels.
From North America: Direct flights from Boston and Toronto; seasonally from Oakland and Montreal. They are mainly run by Azores Airlines, who promote the destination as a stopover en route to Europe.
From island to island: local carrier SATA provides inter-island flights to all 9 islands, though these are quite pricey (regardless of distance). Alternatively you can take ferries - see below.
By ferry: Atlanticoline sails between all 9 islands, but we only really recommend ferries within the central island group of Terceira, Sao Jorge, Pico and Faial as these routes are shorter (1-2 hours, or only 30 mins from Faial to Pico), year-round, and less liable to weather disruption. The longer routes tend to tun only from May-September, and can take anything from 4 hours to overnight - so we advise flying these routes (see above).
On each island: you'll almost certainly want to hire a car in order to explore freely: they are not expensive (standard European prices), but stock is limited (and ageing) so you should book well ahead for summer dates. Otherwise, if you're staying somewhere central such as Ponta Delgada, you could use minibus tours, which are widely available and generally well run.