We brought 3 month old twins to Fuerteventura last September, and it's well doable to take one. There are a load of tips to make it a bit easier.
If not breastfeeding, use a 5 litre water bottle for sterilising with Milton tablets it saves immesurably on space. I'd also use bottled water rather than tap water for feeding (choose a low salt one).
Feed on take off and landing as it helps with the ears.
Bring your own sheets for the cot.
Insect repellant is vital - first time round with our (now) 4 year old she got eaten aalive the first night, and came out in spots. She was fine, but to the parents in the next apartment it looked like chickenpox so they studiously avoided us for the week.
Don't use too much suncream if it's very hot - they sweat, and dehydration could be a problem. It's best to have a pram with a good cover to keep the sum of. You can take turns, and just cream up the limbs, obviously keeping them in during the hottest part of the day.
Aside from that, go for it. Fuerteventura might be a bit hot, but the Costa del Sol should be fine. Benalmadena (near the Marina) is nice, and very child friendly.