London Underground

You need to get an Oyster Card (or have a contactless credit/debit card that works in the UK). For only 4 days you are better with daily maximum charges than a weekly pass (the breakeven point is 5 days). Acton Vale is not a station but an area and it is at the boundary between zones 2 and 3. If you are using the tube (and buses) the daily maximum charge from Zone 2 will be £6.40 and from Zone 3 will be £7.50, if you are only using buses then the daily maximum charge will be £4.40. From Heathrow to Shepherds Bush will cost you £1.50 off-peak or £2.80 peak.

You could walk it in 20 minutes or so. https://goo.gl/maps/uAYmX

Failing that, a 207 bus will take you from Shepherd's Bush along Uxbridge Road through Acton Vale - it's a very high frequency service; every 5 minutes during much of the day. To use it you need an oyster card (get it from any tube station (e.g. Heathrow) and it's £1.50 flat fare with a daily cap of £4.40.

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