Weekend travel in Cinque Terre

The main attraction is practically free--hiking the trails.
I would go to Booking.com now and scope out the accommodations options for your date and your specific needs.

Book ahead of time if you want to stay in those villages.
Monterosso has the most options.
You might be able to find something cheaper in the villages northwest of Cinque Terre, like Levanto, Bonassola, even Moneglia. They are just minutes away by train.

Here's the website for the park (the region is essentially an Italian National park) which includes hiking information and which trails require a CT Trekking pass ($). A combo Trekking and Treno pass is also available.
https://www.parconazionale5terre.it/Eindex.php

Interactive Park Trail map and more trail details: (from the website)
https://www.parconazionale5terre.it/Esentieri-outdoor.php
https://www.parconazionale5terre.it/Erete-sentieristica.php

Info about the passes:
https://www.parconazionale5terre.it/Ecinque-terre-card.php

Useful FAQs:
https://www.parconazionale5terre.it/page.php?id=346

Really, the main activities are hiking (when it's not raining), swimming (warm months only) and exploring the 5 little (and they ARE little) villages. Kayak rental is also available. Be aware that some beach areas in Monterosso are private and have a cost to use. This blog (not mine) has a good overview:
https://www.alongdustyroads.com/posts/monterosso-al-mare-cinque-terre
More on places to swim in the CT:
https://www.alongdustyroads.com/posts/best-cinque-terre-beaches

That blog has some other good info to browse as well.
"Budget friendly" accommodations in summer? This isn't an inexpensive place to spend a weekend, and it can be brutally crowded. If rates are too high, look at Levanto: it's just a few minutes by train from Monterosso, and also has a beach.