modernise an old site

Hi
With my very limited skills, I donated this working site site and now it is time to make it a Responsive web design.
What do you suggest?
I am also looking for a hosting company "better than what I now have".
current needs:
20GB space
0.2 GB bandwidth per month

the current set up is:
Joomla version 1.15.15
SermonSpeaker version: I don't know but here is the copy/paste from the bottom of the help page;
"
4: Upgrades
Before doing anything, it is strongly recommended to create a backup of your files and database. Check out JoomlaPack if you don't already have a backup tool.
Since Version 3.3, SermonSpeaker and its modules support upgrading - there are no more upgrade packages available. Simple install the new version, the database and all files are upgraded automatically. After upgrading, go to the configuration page and check out the new features. It is important that you save the config to update the config files (if you don't save it, SermonSpeaker maybe produce errors!)
"

Do I need to use joomla still? Is there a freelancer which can help me out? I found a new hosting company "I hope I am making the right move" chemicloud.com web hosting pro $5.95 / month which will be also used for another web site I am moving.

I can't help you with choosing a new hosting company, but I can guide you with the rest a bit.
as you know, Joomla 1.5 as well as SermonSpeaker 3.4 (the one you have installed) are outdated by many years. So updating the software is a must.
SermonSpeaker only works with Joomla. If you want to build your site with another CMS, then you'll need to migrate the data somehow, which is likely more work than just upgrading Joomla and SermonSpeaker

So now that is put aside, have a look at the migrate article I wrote seven years ago: www.sermonspeaker.net/faq/27-how-do-i-mi...rmonspeaker-4-x.html
If you follow that guide, it should work.

So the new web hosting will have the latest joomla and I can install the latest SeromnSpeaker, then re-enter all the data from the older web site again which few hundred entries.
Is there an easier way to do it?

i.e.
since I am doing away with the old web hosting, why wast time upgrading?

Any suggestions?
thx

It is not possible to directly upgrade a Joomla 1.5 site anyway. You always need a second installation and migrate the data.
The article I linked above explains how to migrate the SermonSpeaker data.
Tools like JUpgrade help you with migrating the Joomla Core data (eg articles, contacts).

For the SermonSpeaker data, please follow the exact steps written in that article. Most importantly don't install the latest SermonSpeaker version before it tells you to do so.

After installing the latest Joomla on a new server (no upgrade here), I will be left with 2 servers;
1) Old server running Joomla 1.15.15 with SermonSpeaker 3.4
2) New server running Joomla 3.9.8

now to the SermonSpeaker, what is the easiest way:
Do I have options? i.e.
1) install the old 3.4 version to the new Joomla/server, then do the upgrade following your link?
2) install the new version and re create all the sermons titles "too much writing".
3) do I have any other options?

Thanks again

Did you read the article I linked? I've written there exactly what you need to do. I'm not going to repeat what is written there.