Advice on Updating Existing System

I have an existing hard wired system in my home. I haven't tinkered with the system much because I know very little about security systems. I do my own home automation for all other areas of the house. However, I would like to solve a few problems and get some remote access to the system.

The system is currently monitored. It's an older Honeywell panel. There are 2 keypads and 16 wired zones. It is connected to an IP landline.

The problems have been: (1) I don't like the company monitoring the system. Bad customer service and expensive. (2) The connection to the IP phone line causes a comm failure 2-3 times a week when the service drops in the middle of the night.

What I'd like to accomplish (1) get the system off the IP landline (2) get a newer panel (installed by a professional) that I can work with myself and add other features (3) install a couple wireless smokes and glass break detectors (4) consider monitoring options. I'm not really thrilled with self-monitoring but I really don't want to get locked in to a 3 year contract (5) get remote access to arm/disarm the system and monitor folks going in and out of the house.

My first thought was to pay someone to install an Elk panel and cellular communicator then let me take it from there. Maybe get a less expensive monitoring service on a month to month basis and try that and self monitoring and see what I like.

I realize this is somewhat vague but any suggestions would be appreciated.

If you want the full features of the Elk, it's certainly a great panel... but it's not required for what you've stated.  There are cheaper panels that also have what you've listed so far;
 
Sounds like regardless you need something with a cellular transmitter and you can get hooked up with alarmrelay or similar.  Professional alarm installers love the revenue they get from those recurring fees - it's hard to find a pro that'll just work by the hour doing what you want them to do - I tried a long time ago.  Honestly installing an alarm isn't all that hard; sure you'll have a little to learn but once you do the concepts are really simple and this forum is a great resource if you get stuck as are the places that sell to DIYers (like automatedoutlet.com).
 
Self monitoring is always frowned upon - because you are the weakest link; there's no way you can be available 100% of the time - if you're on a plane or in a meeting, what will you do?
 
If you do get an Elk, you of course have off-the-shelf remote access programs and it's compatible with the Uplink cellular transmitter, and alarmrelay will monitor it for you...  so it's a nice solution.

Your Internet connect drops in the middle of the night? Why?
 
I had a provider years ago...well I'll mention names... that used Sympatico's line. In order to reduce bandwidth they would send me a router disconnect code (recorded inside the router events) frequently that would cause my crappy old DLink router to reboot. The answer, of course, was "You are disconnecting", "virus?", "DoS", "It's not us", "you imagined it" . I talked to DLink and finally, after digging real hard, got the response that their new routers all do the same thing. I went through hell and back trying to understand all the layers of communication and discovered their other Internet based excuses were all BS. I don't use that copper pair medium anymore and I won't touch another DLunk product again.
 
This was a P2P prevention (you had to reconnect and buildup a peer bank all over again taking about 3-5 minutes) by annoyance and since, in Canada, the CRTC has passed regulations against doing those techniques.
 
I really wonder if cell service is any more reliable. Hell I get emails and SMS messages from my ISY system sometimes hours later or the next day. When some big news event happens cell systems are "gone fishun' ". A copper pair is usually a reliable resource for sure.

Alarm Relay can probably monitor what you have now.  You can call them with the model number and they can confirm or not, and will even help you re-program it for their monitoring service.
 
You didn't mention what kind of automation you already have.  Many of the "Security only" panels such as DSC and GE/Interlogix are supported by a large number of systems out there already.

My home automation is a mix of different things. Belkin switches, Ecobee thermostat, garage door opener, etc.

A local company offered me a GE Concord 4 panel and alarm.com for $39 p/mth. I'm considering that, but I really don't like the 3 yr contract.

I would suggest to get a professional installation service. There are lots of home security providers out there that provides affordable home security packages. What's the best thing about it is the fact that you can get 24/7 professional monitoring, you can just sit back and relax while your system is being installed for you and you can ensure that they can install the system correctly. Just weigh your choices and choose the best options based on your situation.

You need to provide the information about what the panel make/model is first and foremost.
 
Usually, unless you really see a bunch of features that can't be supported by anything but the latest Black I-droid whatever alarm system (remember, alarms are NOT items that keep up with the Joneses) then usually it's a case that you're not going to see much benefit on replacing hardware. Still, unless the panel is completely locked out of programming or 100% proprietary, the need to yank it for something new usually is a lot of effort for little returns.
 
The comms problem is one item that MUST be addressed no matter if you choose to yank the panel or not. You've already know there's an issue with that that isn't going to be fixed by pulling the panel. Even going IP monitoring isn't going to be an absolute as there is a lot that needs to be considered on your end to have a reliable transmission path. Not something I'd recommend for most applications.
 
Cellular monitoring is viable, can even be used in the case the panel can't be reprogrammed and reasonably foolproof compared to IP monitoring. While nothing is 100%, it's a better average and more cost effective than a panel swap in the majority of the cases.
 
From the sounds of your automation, you have a mix-mash of a bunch of standalone items that are most likely unable to be completely integrated no matter what the panel is.