Regrettably, there is no easy nor cheap option to allow you to travel direct from Ibiza to Lagos or the Algarve.
Air Berlin (http://www.airberlin.com) do operate daily flights from Palma da Mallorca to Faro in the Algarve. There do appear to be ferries between Ibiza to Palma,(see for example, http://www.balearia.net)
There are no direct buses from Faro airport to Lagos. You would have to arrange a hire car (I recommend http://www.lagoscarhire.com/), an airport transfer, (e.g. http://www.low-cost-transfers.com/faro-airport/index.php) or take a taxi. There are trains from Faro town to Lagos, but you would have to take a taxi into Faro and then should make sure that you have a taxi waiting for you on arrival at Lagos station as there are often none there on arrival.
As regards the return trip by train, this will be tortuous. You will need to catch a local train from Lagos to Tunes Junction. There you connect to the Alfa-Pendular high speed train to Lisbon Gare Oriente Station. From Gare Oriente, you will need to connect with the daily 16:14 Overnight Sud Express Departure for Hendaye where it connects with a TGV to Paris Montparnasse. From there you will have to transit to Gare du Nord to catch the Eurostar.
Links (pdf format) to the train timetables can be found on my blog on the right hand side in the panel entitled:
>> Algarve & Portugal - Travel Transport Links
You will need to check three timetables:
> Algarve Regional Line Train Timetable
> Porto - Lisbon - Faro (Algarve) Train Timetable
> Paris to Lisbon Train Timetable - Sud Expresso
