If you are just in Olhão for a few days, and happy to potter along the coast then you don’t need a car as there are good public transport links to the Islands, Faro, Tavira and Vila Real de Santo António. We’ve made a few trips by train to the east of Olhão. . .
It is a fun way to travel, you get some great views from the trains and some of the stations are just lovely. So even if you hire a car or have driven from home in your own, we recommend getting the train at least once. However, there is a but, and it is quite a big one at the moment they do not have the funds to make the local trains accessible. The step up from nearly every platform, and of course down again, is enormous. If you have any mobility issues, however slight, you might find it too big a challenge despite the willingness of locals to help. Check out my links page for more information on the buses and trains.
Vila Real de Santo António is on the Rio Guadiana.
It is one of the more touristy towns in the east, mainly due to the influx of the Spanish who cross by ferry or road from Ayamonte in Spain. There is a lovely seafront and also town square, but for some reason it just didn’t appeal to us. We have travelled across to the area since by car to investigate the towns of Castro Marim and Alcoutim, and both these towns and the surrounding countryside definitely warrant repeat visits. I’ll be posting about those visits in due course. One visit to Vila Real de Santo António however was sufficient for us, but I suspect we may be in the minority on this! I am sure others will find it beautifully quaint. It is very flat so perfect for walking around, although do wear comfy shoes as like many towns in the Algarve it is mostly cobbles. One 
of the delights though of travelling to Vila Real de Santo António by train is for the views of Tavira when you cross the viaduct, and to enjoy the stunning countryside all along this part of the Algarve.
Tavira on the other hand, we love. 
It is incredibly touristy compared to the rest of East Algarve, and in some ways almost too picture perfect. We though really enjoy strolling around its small streets, and we have found some wonderful cafes to have lunch in and also to watch the world go past. The pedestrianised bridge, Ponte Romana, spans the river with low arches, is incredibly photographic and at low tide you can watch fisherman, waders and even Egrets forage amongst the rock pools.
One of the lovely things about Tavira is that from the castle you look across the roof tops to the saltpans.
