Nice to read this !
Good to know that someone else likes Elche - I have a real "soft spot" for this town, one of my favourite places in Spain.
From memory I think the town is very much one of two halves; on one side of the river, over the bridge, it's just like any normal Spanish urban area (though not unpleasant by any means) but on the other side it's very attractive ...cathedral, archaeological museum, famous botanical garden....and as said above a very beautiful and very peaceful municipal park. It's a unique place I think -Palm Trees just dominate the town and it makes it so different to other areas you might visit - as a tourist, it's well worth a day trip from the Costa Blanca.
I think there are two train stations and you have to get off at the first (if travelling from Alicante) - "Elche Park". Only about 30-40 mins journey ? Then a short walk to the municipal park, interesting museum & cathedral. The botanic garden is a bit further away but I would say it's within walking distance, well worth a visit.
...By the way, when you enter the botanic garden they take a sneaky photo of you and then when you leave they already have a commemorative key ring made up with your picture in, available for a few euros.
It's a good little fundraising scheme, and hard to turn down !